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+The Project Gutenberg eBook of Relativity: The Special and General Theory, by Albert Einstein
+
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online at
+www.gutenberg.org. If you are not located in the United States, you
+will have to check the laws of the country where you are located before
+using this eBook.
+
+Title: Relativity: The Special and General Theory
+
+Author: Albert Einstein
+
+Release Date: October 1, 2009 [eBook #30155]
+[Most recently updated: May 2, 2023]
+
+Language: English
+
+Produced by: Robert Hux
+
+*** START OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***
+
+
+
+
+cover 
+
+
+
+
+Relativity: The Special and General Theory
+
+by Albert Einstein
+
+
+Authorised Translation by Robert W. Lawson
+
+
+
+
+ALBERT EINSTEIN REFERENCE ARCHIVE
+RELATIVITY: THE SPECIAL AND GENERAL THEORY
+BY ALBERT EINSTEIN
+
+
+Written: 1916 (this revised edition: 1924)
+Source: Relativity: The Special and General Theory (1920)
+Publisher: Methuen & Co Ltd
+First Published: December, 1916
+Translated: Robert W. Lawson (Authorised translation)
+Transcription/Markup: Brian Basgen
+Transcription to text: Gregory B. Newby
+Thanks to: Einstein Reference Archive (marxists.org)
+The Einstein Reference Archive is online at:
+http://www.marxists.org/reference/archive/einstein/index.htm
+
+
+
+
+Contents
+
+ Preface
+
+ Part I: The Special Theory of Relativity
+ I. Physical Meaning of Geometrical Propositions
+ II. The System of Co-ordinates
+ III. Space and Time in Classical Mechanics
+ IV. The Galileian System of Co-ordinates
+ V. The Principle of Relativity (in the Restricted Sense)
+ VI. The Theorem of the Addition of Velocities employed in Classical Mechanics
+ VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity
+ VIII. On the Idea of Time in Physics
+ IX. The Relativity of Simultaneity
+ X. On the Relativity of the Conception of Distance
+ XI. The Lorentz Transformation
+ XII. The Behaviour of Measuring-Rods and Clocks in Motion
+ XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau
+ XIV. The Heuristic Value of the Theory of Relativity
+ XV. General Results of the Theory
+ XVI. Experience and the Special Theory of Relativity
+ XVII. Minkowski’s Four-dimensional Space
+
+ Part II: The General Theory of Relativity
+ XVIII. Special and General Principle of Relativity
+ XIX. The Gravitational Field
+ XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
+ XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
+ XXII. A Few Inferences from the General Principle of Relativity
+ XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
+ XXIV. Euclidean and non-Euclidean Continuum
+ XXV. Gaussian Co-ordinates
+ XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
+ XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum
+ XXVIII. Exact Formulation of the General Principle of Relativity
+ XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
+
+ Part III: Considerations on the Universe as a Whole
+ XXX. Cosmological Difficulties of Newton’s Theory
+ XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe
+ XXXII. The Structure of Space According to the General Theory of Relativity
+
+ Appendices:
+ I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)
+ II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII)
+ III. The Experimental Confirmation of the General Theory of Relativity
+ IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII)
+ V. Relativity and the Problem of Space
+
+
+Note: The fifth Appendix was added by Einstein at the time of the
+fifteenth re-printing of this book; and as a result is still under
+copyright restrictions so cannot be added without the permission of the
+publisher.
+
+
+
+
+PREFACE
+
+
+The present book is intended, as far as possible, to give an exact
+insight into the theory of Relativity to those readers who, from a
+general scientific and philosophical point of view, are interested in
+the theory, but who are not conversant with the mathematical apparatus
+of theoretical physics. The work presumes a standard of education
+corresponding to that of a university matriculation examination, and,
+despite the shortness of the book, a fair amount of patience and force
+of will on the part of the reader. The author has spared himself no
+pains in his endeavour to present the main ideas in the simplest and
+most intelligible form, and on the whole, in the sequence and
+connection in which they actually originated. In the interest of
+clearness, it appeared to me inevitable that I should repeat myself
+frequently, without paying the slightest attention to the elegance of
+the presentation. I adhered scrupulously to the precept of that
+brilliant theoretical physicist L. Boltzmann, according to whom matters
+of elegance ought to be left to the tailor and to the cobbler. I make
+no pretence of having withheld from the reader difficulties which are
+inherent to the subject. On the other hand, I have purposely treated
+the empirical physical foundations of the theory in a “step-motherly”
+fashion, so that readers unfamiliar with physics may not feel like the
+wanderer who was unable to see the forest for the trees. May the book
+bring some one a few happy hours of suggestive thought!
+
+December, 1916
+
+ A. EINSTEIN
+
+
+
+
+PART I: THE SPECIAL THEORY OF RELATIVITY
+
+
+
+
+I.
+PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
+
+
+In your schooldays most of you who read this book made acquaintance
+with the noble building of Euclid’s geometry, and you remember—perhaps
+with more respect than love—the magnificent structure, on the lofty
+staircase of which you were chased about for uncounted hours by
+conscientious teachers. By reason of our past experience, you would
+certainly regard everyone with disdain who should pronounce even the
+most out-of-the-way proposition of this science to be untrue. But
+perhaps this feeling of proud certainty would leave you immediately if
+some one were to ask you: “What, then, do you mean by the assertion
+that these propositions are true?” Let us proceed to give this question
+a little consideration.
+
+Geometry sets out from certain conceptions such as “plane,” “point,”
+and “straight line,” with which we are able to associate more or less
+definite ideas, and from certain simple propositions (axioms) which, in
+virtue of these ideas, we are inclined to accept as “true.” Then, on
+the basis of a logical process, the justification of which we feel
+ourselves compelled to admit, all remaining propositions are shown to
+follow from those axioms, _i.e._ they are proven. A proposition is then
+correct (“true”) when it has been derived in the recognised manner from
+the axioms. The question of “truth” of the individual geometrical
+propositions is thus reduced to one of the “truth” of the axioms. Now
+it has long been known that the last question is not only unanswerable
+by the methods of geometry, but that it is in itself entirely without
+meaning. We cannot ask whether it is true that only one straight line
+goes through two points. We can only say that Euclidean geometry deals
+with things called “straight lines,” to each of which is ascribed the
+property of being uniquely determined by two points situated on it. The
+concept “true” does not tally with the assertions of pure geometry,
+because by the word “true” we are eventually in the habit of
+designating always the correspondence with a “real” object; geometry,
+however, is not concerned with the relation of the ideas involved in it
+to objects of experience, but only with the logical connection of these
+ideas among themselves.
+
+It is not difficult to understand why, in spite of this, we feel
+constrained to call the propositions of geometry “true.” Geometrical
+ideas correspond to more or less exact objects in nature, and these
+last are undoubtedly the exclusive cause of the genesis of those ideas.
+Geometry ought to refrain from such a course, in order to give to its
+structure the largest possible logical unity. The practice, for
+example, of seeing in a “distance” two marked positions on a
+practically rigid body is something which is lodged deeply in our habit
+of thought. We are accustomed further to regard three points as being
+situated on a straight line, if their apparent positions can be made to
+coincide for observation with one eye, under suitable choice of our
+place of observation.
+
+If, in pursuance of our habit of thought, we now supplement the
+propositions of Euclidean geometry by the single proposition that two
+points on a practically rigid body always correspond to the same
+distance (line-interval), independently of any changes in position to
+which we may subject the body, the propositions of Euclidean geometry
+then resolve themselves into propositions on the possible relative
+position of practically rigid bodies.[1] Geometry which has been
+supplemented in this way is then to be treated as a branch of physics.
+We can now legitimately ask as to the “truth” of geometrical
+propositions interpreted in this way, since we are justified in asking
+whether these propositions are satisfied for those real things we have
+associated with the geometrical ideas. In less exact terms we can
+express this by saying that by the “truth” of a geometrical proposition
+in this sense we understand its validity for a construction with rule
+and compasses.
+
+
+ [1] It follows that a natural object is associated also with a
+ straight line. Three points _A, B_ and _C_ on a rigid body thus lie in
+ a straight line when the points _A_ and _C_ being given, _B_ is chosen
+ such that the sum of the distances _AB_ and _BC_ is as short as
+ possible. This incomplete suggestion will suffice for the present
+ purpose.
+
+
+Of course the conviction of the “truth” of geometrical propositions in
+this sense is founded exclusively on rather incomplete experience. For
+the present we shall assume the “truth” of the geometrical
+propositions, then at a later stage (in the general theory of
+relativity) we shall see that this “truth” is limited, and we shall
+consider the extent of its limitation.
+
+
+
+
+II.
+THE SYSTEM OF CO-ORDINATES
+
+
+On the basis of the physical interpretation of distance which has been
+indicated, we are also in a position to establish the distance between
+two points on a rigid body by means of measurements. For this purpose
+we require a “distance” (rod _S_) which is to be used once and for all,
+and which we employ as a standard measure. If, now, _A_ and _B_ are two
+points on a rigid body, we can construct the line joining them
+according to the rules of geometry; then, starting from _A_, we can
+mark off the distance _S_ time after time until we reach _B_. The
+number of these operations required is the numerical measure of the
+distance _AB_. This is the basis of all measurement of length.[2]
+
+
+ [2] Here we have assumed that there is nothing left over _i.e._ that
+ the measurement gives a whole number. This difficulty is got over by
+ the use of divided measuring-rods, the introduction of which does not
+ demand any fundamentally new method.
+
+
+Every description of the scene of an event or of the position of an
+object in space is based on the specification of the point on a rigid
+body (body of reference) with which that event or object coincides.
+This applies not only to scientific description, but also to everyday
+life. If I analyse the place specification “Trafalgar Square,
+London”[3] I arrive at the following result. The earth is the rigid
+body to which the specification of place refers; “Trafalgar Square,
+London” is a well-defined point, to which a name has been assigned, and
+with which the event coincides in space.[4]
+
+
+ [3]
+
+I have chosen this as being more familiar to the English reader than
+the “Potzdammer Platz, Berlin,” which is referred to in the original.
+(R. W. L.)
+
+
+ [4] It is not necessary here to investigate further the significance
+ of the expression “coincidence in space.” This conception is
+ sufficiently obvious to ensure that differences of opinion are
+ scarcely likely to arise as to its applicability in practice.
+
+
+This primitive method of place specification deals only with places on
+the surface of rigid bodies, and is dependent on the existence of
+points on this surface which are distinguishable from each other. But
+we can free ourselves from both of these limitations without altering
+the nature of our specification of position. If, for instance, a cloud
+is hovering over Trafalgar Square, then we can determine its position
+relative to the surface of the earth by erecting a pole perpendicularly
+on the Square, so that it reaches the cloud. The length of the pole
+measured with the standard measuring-rod, combined with the
+specification of the position of the foot of the pole, supplies us with
+a complete place specification. On the basis of this illustration, we
+are able to see the manner in which a refinement of the conception of
+position has been developed.
+
+(_a_) We imagine the rigid body, to which the place specification is
+referred, supplemented in such a manner that the object whose position
+we require is reached by the completed rigid body.
+
+(_b_) In locating the position of the object, we make use of a number
+(here the length of the pole measured with the measuring-rod) instead
+of designated points of reference.
+
+(_c_) We speak of the height of the cloud even when the pole which
+reaches the cloud has not been erected. By means of optical
+observations of the cloud from different positions on the ground, and
+taking into account the properties of the propagation of light, we
+determine the length of the pole we should have required in order to
+reach the cloud.
+
+From this consideration we see that it will be advantageous if, in the
+description of position, it should be possible by means of numerical
+measures to make ourselves independent of the existence of marked
+positions (possessing names) on the rigid body of reference. In the
+physics of measurement this is attained by the application of the
+Cartesian system of co-ordinates.
+
+This consists of three plane surfaces perpendicular to each other and
+rigidly attached to a rigid body. Referred to a system of co-ordinates,
+the scene of any event will be determined (for the main part) by the
+specification of the lengths of the three perpendiculars or
+co-ordinates (_x, y, z_) which can be dropped from the scene of the
+event to those three plane surfaces. The lengths of these three
+perpendiculars can be determined by a series of manipulations with
+rigid measuring-rods performed according to the rules and methods laid
+down by Euclidean geometry.
+
+In practice, the rigid surfaces which constitute the system of
+co-ordinates are generally not available; furthermore, the magnitudes
+of the co-ordinates are not actually determined by constructions with
+rigid rods, but by indirect means. If the results of physics and
+astronomy are to maintain their clearness, the physical meaning of
+specifications of position must always be sought in accordance with the
+above considerations.[5]
+
+
+ [5] A refinement and modification of these views does not become
+ necessary until we come to deal with the general theory of relativity,
+ treated in the second part of this book.
+
+
+We thus obtain the following result: Every description of events in
+space involves the use of a rigid body to which such events have to be
+referred. The resulting relationship takes for granted that the laws of
+Euclidean geometry hold for “distances;” the “distance” being
+represented physically by means of the convention of two marks on a
+rigid body.
+
+
+III.
+
+SPACE AND TIME IN CLASSICAL MECHANICS
+
+The purpose of mechanics is to describe how bodies change their
+position in space with “time.” I should load my conscience with grave
+sins against the sacred spirit of lucidity were I to formulate the aims
+of mechanics in this way, without serious reflection and detailed
+explanations. Let us proceed to disclose these sins.
+
+It is not clear what is to be understood here by “position” and
+“space.” I stand at the window of a railway carriage which is
+travelling uniformly, and drop a stone on the embankment, without
+throwing it. Then, disregarding the influence of the air resistance, I
+see the stone descend in a straight line. A pedestrian who observes the
+misdeed from the footpath notices that the stone falls to earth in a
+parabolic curve. I now ask: Do the “positions” traversed by the stone
+lie “in reality” on a straight line or on a parabola? Moreover, what is
+meant here by motion “in space”? From the considerations of the
+previous section the answer is self-evident. In the first place we
+entirely shun the vague word “space,” of which, we must honestly
+acknowledge, we cannot form the slightest conception, and we replace it
+by “motion relative to a practically rigid body of reference.” The
+positions relative to the body of reference (railway carriage or
+embankment) have already been defined in detail in the preceding
+section. If instead of “body of reference” we insert “system of
+co-ordinates,” which is a useful idea for mathematical description, we
+are in a position to say: The stone traverses a straight line relative
+to a system of co-ordinates rigidly attached to the carriage, but
+relative to a system of co-ordinates rigidly attached to the ground
+(embankment) it describes a parabola. With the aid of this example it
+is clearly seen that there is no such thing as an independently
+existing trajectory (lit. “path-curve”[6], but only a trajectory
+relative to a particular body of reference.
+
+
+ [6] That is, a curve along which the body moves.
+
+
+In order to have a _complete_ description of the motion, we must
+specify how the body alters its position _with time; i.e._ for every
+point on the trajectory it must be stated at what time the body is
+situated there. These data must be supplemented by such a definition of
+time that, in virtue of this definition, these time-values can be
+regarded essentially as magnitudes (results of measurements) capable of
+observation. If we take our stand on the ground of classical mechanics,
+we can satisfy this requirement for our illustration in the following
+manner. We imagine two clocks of identical construction; the man at the
+railway-carriage window is holding one of them, and the man on the
+footpath the other. Each of the observers determines the position on
+his own reference-body occupied by the stone at each tick of the clock
+he is holding in his hand. In this connection we have not taken account
+of the inaccuracy involved by the finiteness of the velocity of
+propagation of light. With this and with a second difficulty prevailing
+here we shall have to deal in detail later.
+
+
+IV. THE GALILEIAN SYSTEM OF CO-ORDINATES
+
+
+As is well known, the fundamental law of the mechanics of
+Galilei-Newton, which is known as the _law of inertia_, can be stated
+thus: A body removed sufficiently far from other bodies continues in a
+state of rest or of uniform motion in a straight line. This law not
+only says something about the motion of the bodies, but it also
+indicates the reference-bodies or systems of coordinates, permissible
+in mechanics, which can be used in mechanical description. The visible
+fixed stars are bodies for which the law of inertia certainly holds to
+a high degree of approximation. Now if we use a system of co-ordinates
+which is rigidly attached to the earth, then, relative to this system,
+every fixed star describes a circle of immense radius in the course of
+an astronomical day, a result which is opposed to the statement of the
+law of inertia. So that if we adhere to this law we must refer these
+motions only to systems of coordinates relative to which the fixed
+stars do not move in a circle. A system of co-ordinates of which the
+state of motion is such that the law of inertia holds relative to it is
+called a “Galileian system of co-ordinates.” The laws of the mechanics
+of Galilei-Newton can be regarded as valid only for a Galileian system
+of co-ordinates.
+
+
+V.
+
+THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)
+
+In order to attain the greatest possible clearness, let us return to
+our example of the railway carriage supposed to be travelling
+uniformly. We call its motion a uniform translation (“uniform” because
+it is of constant velocity and direction, “translation” because
+although the carriage changes its position relative to the embankment
+yet it does not rotate in so doing). Let us imagine a raven flying
+through the air in such a manner that its motion, as observed from the
+embankment, is uniform and in a straight line. If we were to observe
+the flying raven from the moving railway carriage. we should find that
+the motion of the raven would be one of different velocity and
+direction, but that it would still be uniform and in a straight line.
+Expressed in an abstract manner we may say: If a mass _m_ is moving
+uniformly in a straight line with respect to a co-ordinate system _K_,
+then it will also be moving uniformly and in a straight line relative
+to a second co-ordinate system _K′_ provided that the latter is
+executing a uniform translatory motion with respect to _K_. In
+accordance with the discussion contained in the preceding section, it
+follows that:
+
+If _K_ is a Galileian co-ordinate system. then every other co-ordinate
+system _K′_ is a Galileian one, when, in relation to _K_, it is in a
+condition of uniform motion of translation. Relative to _K′_ the
+mechanical laws of Galilei-Newton hold good exactly as they do with
+respect to _K_.
+
+We advance a step farther in our generalisation when we express the
+tenet thus: If, relative to _K_, _K′_ is a uniformly moving co-ordinate
+system devoid of rotation, then natural phenomena run their course with
+respect to _K′_ according to exactly the same general laws as with
+respect to _K_. This statement is called the _principle of relativity_
+(in the restricted sense).
+
+As long as one was convinced that all natural phenomena were capable of
+representation with the help of classical mechanics, there was no need
+to doubt the validity of this principle of relativity. But in view of
+the more recent development of electrodynamics and optics it became
+more and more evident that classical mechanics affords an insufficient
+foundation for the physical description of all natural phenomena. At
+this juncture the question of the validity of the principle of
+relativity became ripe for discussion, and it did not appear impossible
+that the answer to this question might be in the negative.
+
+Nevertheless, there are two general facts which at the outset speak
+very much in favour of the validity of the principle of relativity.
+Even though classical mechanics does not supply us with a sufficiently
+broad basis for the theoretical presentation of all physical phenomena,
+still we must grant it a considerable measure of “truth,” since it
+supplies us with the actual motions of the heavenly bodies with a
+delicacy of detail little short of wonderful. The principle of
+relativity must therefore apply with great accuracy in the domain of
+_mechanics_. But that a principle of such broad generality should hold
+with such exactness in one domain of phenomena, and yet should be
+invalid for another, is _a priori_ not very probable.
+
+We now proceed to the second argument, to which, moreover, we shall
+return later. If the principle of relativity (in the restricted sense)
+does not hold, then the Galileian co-ordinate systems _K, K′, K″_,
+etc., which are moving uniformly relative to each other, will not be
+_equivalent_ for the description of natural phenomena. In this case we
+should be constrained to believe that natural laws are capable of being
+formulated in a particularly simple manner, and of course only on
+condition that, from amongst all possible Galileian co-ordinate
+systems, we should have chosen _one_ (_K0_) of a particular state of
+motion as our body of reference. We should then be justified (because
+of its merits for the description of natural phenomena) in calling this
+system “absolutely at rest,” and all other Galileian systems _K_ “in
+motion.” If, for instance, our embankment were the system _K0_ then our
+railway carriage would be a system _K_, relative to which less simple
+laws would hold than with respect to _K0_. This diminished simplicity
+would be due to the fact that the carriage _K_ would be in motion
+(_i.e._ “really”)with respect to _K0_. In the general laws of nature
+which have been formulated with reference to _K_, the magnitude and
+direction of the velocity of the carriage would necessarily play a
+part. We should expect, for instance, that the note emitted by an
+organpipe placed with its axis parallel to the direction of travel
+would be different from that emitted if the axis of the pipe were
+placed perpendicular to this direction.
+
+Now in virtue of its motion in an orbit round the sun, our earth is
+comparable with a railway carriage travelling with a velocity of about
+30 kilometres per second. If the principle of relativity were not valid
+we should therefore expect that the direction of motion of the earth at
+any moment would enter into the laws of nature, and also that physical
+systems in their behaviour would be dependent on the orientation in
+space with respect to the earth. For owing to the alteration in
+direction of the velocity of revolution of the earth in the course of a
+year, the earth cannot be at rest relative to the hypothetical system
+_K0_ throughout the whole year. However, the most careful observations
+have never revealed such anisotropic properties in terrestrial physical
+space, _i.e._ a physical non-equivalence of different directions. This
+is very powerful argument in favour of the principle of relativity.
+
+
+VI.
+
+THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL
+MECHANICS
+
+Let us suppose our old friend the railway carriage to be travelling
+along the rails with a constant velocity _v_, and that a man traverses
+the length of the carriage in the direction of travel with a velocity
+_w_. How quickly or, in other words, with what velocity _W_ does the
+man advance relative to the embankment during the process? The only
+possible answer seems to result from the following consideration: If
+the man were to stand still for a second, he would advance relative to
+the embankment through a distance _v_ equal numerically to the velocity
+of the carriage. As a consequence of his walking, however, he traverses
+an additional distance w relative to the carriage, and hence also
+relative to the embankment, in this second, the distance w being
+numerically equal to the velocity with which he is walking. Thus in
+total he covers the distance _W = v + w_ relative to the embankment in
+the second considered. We shall see later that this result, which
+expresses the theorem of the addition of velocities employed in
+classical mechanics, cannot be maintained; in other words, the law that
+we have just written down does not hold in reality. For the time being,
+however, we shall assume its correctness.
+
+
+VII.
+
+THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH
+THE PRINCIPLE OF RELATIVITY
+
+There is hardly a simpler law in physics than that according to which
+light is propagated in empty space. Every child at school knows, or
+believes he knows, that this propagation takes place in straight lines
+with a velocity _c_ = 300,000 km./sec. At all events we know with great
+exactness that this velocity is the same for all colours, because if
+this were not the case, the minimum of emission would not be observed
+simultaneously for different colours during the eclipse of a fixed star
+by its dark neighbour. By means of similar considerations based on
+observations of double stars, the Dutch astronomer De Sitter was also
+able to show that the velocity of propagation of light cannot depend on
+the velocity of motion of the body emitting the light. The assumption
+that this velocity of propagation is dependent on the direction “in
+space” is in itself improbable.
+
+In short, let us assume that the simple law of the constancy of the
+velocity of light _c_ (in vacuum) is justifiably believed by the child
+at school. Who would imagine that this simple law has plunged the
+conscientiously thoughtful physicist into the greatest intellectual
+difficulties? Let us consider how these difficulties arise.
+
+Of course we must refer the process of the propagation of light (and
+indeed every other process) to a rigid reference-body (co-ordinate
+system). As such a system let us again choose our embankment. We shall
+imagine the air above it to have been removed. If a ray of light be
+sent along the embankment, we see from the above that the tip of the
+ray will be transmitted with the velocity _c_ relative to the
+embankment. Now let us suppose that our railway carriage is again
+travelling along the railway lines with the velocity _v_, and that its
+direction is the same as that of the ray of light, but its velocity of
+course much less. Let us inquire about the velocity of propagation of
+the ray of light relative to the carriage. It is obvious that we can
+here apply the consideration of the previous section, since the ray of
+light plays the part of the man walking along relatively to the
+carriage. The velocity _W_ of the man relative to the embankment is
+here replaced by the velocity of light relative to the embankment. _w_
+is the required velocity of light with respect to the carriage, and we
+have
+
+_w = c – v._
+
+The velocity of propagation ot a ray of light relative to the carriage
+thus comes out smaller than _c_.
+
+But this result comes into conflict with the principle of relativity
+set forth in Section V. For, like every other general law of nature,
+the law of the transmission of light _in vacuo_ [in vacuum] must,
+according to the principle of relativity, be the same for the railway
+carriage as reference-body as when the rails are the body of reference.
+But, from our above consideration, this would appear to be impossible.
+If every ray of light is propagated relative to the embankment with the
+velocity _c_, then for this reason it would appear that another law of
+propagation of light must necessarily hold with respect to the
+carriage—a result contradictory to the principle of relativity.
+
+In view of this dilemma there appears to be nothing else for it than to
+abandon either the principle of relativity or the simple law of the
+propagation of light _in vacuo_. Those of you who have carefully
+followed the preceding discussion are almost sure to expect that we
+should retain the principle of relativity, which appeals so
+convincingly to the intellect because it is so natural and simple. The
+law of the propagation of light _in vacuo_ would then have to be
+replaced by a more complicated law conformable to the principle of
+relativity. The development of theoretical physics shows, however, that
+we cannot pursue this course. The epoch-making theoretical
+investigations of H. A. Lorentz on the electrodynamical and optical
+phenomena connected with moving bodies show that experience in this
+domain leads conclusively to a theory of electromagnetic phenomena, of
+which the law of the constancy of the velocity of light in vacuo is a
+necessary consequence. Prominent theoretical physicists were therefore
+more inclined to reject the principle of relativity, in spite of the
+fact that no empirical data had been found which were contradictory to
+this principle.
+
+At this juncture the theory of relativity entered the arena. As a
+result of an analysis of the physical conceptions of time and space, it
+became evident that _in reality there is not the least incompatibilitiy
+between the principle of relativity and the law of propagation of
+light_, and that by systematically holding fast to both these laws a
+logically rigid theory could be arrived at. This theory has been called
+the _special theory of relativity_ to distinguish it from the extended
+theory, with which we shall deal later. In the following pages we shall
+present the fundamental ideas of the special theory of relativity.
+
+
+VIII.
+
+ON THE IDEA OF TIME IN PHYSICS
+
+Lightning has struck the rails on our railway embankment at two places
+_A_ and _B_ far distant from each other. I make the additional
+assertion that these two lightning flashes occurred simultaneously. If
+I ask you whether there is sense in this statement, you will answer my
+question with a decided “Yes.” But if I now approach you with the
+request to explain to me the sense of the statement more precisely, you
+find after some consideration that the answer to this question is not
+so easy as it appears at first sight.
+
+After some time perhaps the following answer would occur to you: “The
+significance of the statement is clear in itself and needs no further
+explanation; of course it would require some consideration if I were to
+be commissioned to determine by observations whether in the actual case
+the two events took place simultaneously or not.” I cannot be satisfied
+with this answer for the following reason. Supposing that as a result
+of ingenious considerations an able meteorologist were to discover that
+the lightning must always strike the places _A_ and _B_ simultaneously,
+then we should be faced with the task of testing whether or not this
+theoretical result is in accordance with the reality. We encounter the
+same difficulty with all physical statements in which the conception
+“simultaneous” plays a part. The concept does not exist for the
+physicist until he has the possibility of discovering whether or not it
+is fulfilled in an actual case. We thus require a definition of
+simultaneity such that this definition supplies us with the method by
+means of which, in the present case, he can decide by experiment
+whether or not both the lightning strokes occurred simultaneously. As
+long as this requirement is not satisfied, I allow myself to be
+deceived as a physicist (and of course the same applies if I am not a
+physicist), when I imagine that I am able to attach a meaning to the
+statement of simultaneity. (I would ask the reader not to proceed
+farther until he is fully convinced on this point.)
+
+After thinking the matter over for some time you then offer the
+following suggestion with which to test simultaneity. By measuring
+along the rails, the connecting line _AB_ should be measured up and an
+observer placed at the mid-point M of the distance _AB_. This observer
+should be supplied with an arrangement (_e.g._ two mirrors inclined at
+90°) which allows him visually to observe both places _A_ and _B_ at
+the same time. If the observer perceives the two flashes of lightning
+at the same time, then they are simultaneous.
+
+I am very pleased with this suggestion, but for all that I cannot
+regard the matter as quite settled, because I feel constrained to raise
+the following objection: “Your definition would certainly be right, if
+only I knew that the light by means of which the observer at _M_
+perceives the lightning flashes travels along the length _A_ → _M_ with
+the same velocity as along the length _B_ → _M_. But an examination of
+this supposition would only be possible if we already had at our
+disposal the means of measuring time. It would thus appear as though we
+were moving here in a logical circle.”
+
+After further consideration you cast a somewhat disdainful glance at
+me—and rightly so—and you declare: “I maintain my previous definition
+nevertheless, because in reality it assumes absolutely nothing about
+light. There is only _one_ demand to be made of the definition of
+simultaneity, namely, that in every real case it must supply us with an
+empirical decision as to whether or not the conception that has to be
+defined is fulfilled. That my definition satisfies this demand is
+indisputable. That light requires the same time to traverse the path
+_A_ → _M_ as for the path _B_ → _M_ is in reality neither a
+_supposition nor a hypothesis_ about the physical nature of light, but
+a _stipulation_ which I can make of my own freewill in order to arrive
+at a definition of simultaneity.”
+
+It is clear that this definition can be used to give an exact meaning
+not only to _two_ events, but to as many events as we care to choose,
+and independently of the positions of the scenes of the events with
+respect to the body of reference[7] (here the railway embankment). We
+are thus led also to a definition of “time” in physics. For this
+purpose we suppose that clocks of identical construction are placed at
+the points _A, B_ and _C_ of the railway line (co-ordinate system) and
+that they are set in such a manner that the positions of their pointers
+are simultaneously (in the above sense) the same. Under these
+conditions we understand by the “time” of an event the reading
+(position of the hands) of that one of these clocks which is in the
+immediate vicinity (in space) of the event. In this manner a time-value
+is associated with every event which is essentially capable of
+observation.
+
+
+ [7] We suppose further that, when three events _A, B_ and _C_ occur in
+ different places in such a manner that, if _A_ is simultaneous with
+ _B_, and _B_ is simultaneous with _C_ (simultaneous in the sense of
+ the above definition), then the criterion for the simultaneity of the
+ pair of events _A, C_ is also satisfied. This assumption is a physical
+ hypothesis about the law of propagation of light; it must certainly be
+ fulfilled if we are to maintain the law of the constancy of the
+ velocity of light _in vacuo_.
+
+
+This stipulation contains a further physical hypothesis, the validity
+of which will hardly be doubted without empirical evidence to the
+contrary. It has been assumed that all these clocks _go at the same
+rate_ if they are of identical construction. Stated more exactly: When
+two clocks arranged at rest in different places of a reference-body are
+set in such a manner that a _particular_ position of the pointers of
+the one clock is _simultaneous_ (in the above sense) with the _same_
+position, of the pointers of the other clock, then identical “settings”
+are always simultaneous (in the sense of the above definition).
+
+
+IX.
+
+THE RELATIVITY OF SIMULTANEITY
+
+Up to now our considerations have been referred to a particular body of
+reference, which we have styled a “railway embankment.” We suppose a
+very long train travelling along the rails with the constant velocity v
+and in the direction indicated in Fig 1. People travelling in this
+train will with a vantage view the train as a rigid reference-body
+(co-ordinate system); they regard all events in reference to the train.
+Then every event which takes place along the line also takes place at a
+particular point of the train. Also the definition of simultaneity can
+be given relative to the train in exactly the same way as with respect
+to the embankment. As a natural consequence, however, the following
+question arises:
+
+image001
+
+
+Are two events (_e.g._ the two strokes of lightning _A_ and _B_) which
+are simultaneous _with reference to the railway embankment_ also
+simultaneous _relatively to the train?_ We shall show directly that the
+answer must be in the negative.
+
+When we say that the lightning strokes _A_ and _B_ are simultaneous
+with respect to be embankment, we mean: the rays of light emitted at
+the places _A_ and _B_, where the lightning occurs, meet each other at
+the mid-point _M_ of the length _A_ → _B_ of the embankment. But the
+events _A_ and _B_ also correspond to positions _A_ and _B_ on the
+train. Let _M′_ be the mid-point of the distance _A_ → _B_ on the
+travelling train. Just when the flashes (as judged from the embankment)
+of lightning occur, this point _M′_ naturally coincides with the point
+_M_ but it moves towards the right in the diagram with the velocity v
+of the train. If an observer sitting in the position _M′_ in the train
+did not possess this velocity, then he would remain permanently at M,
+and the light rays emitted by the flashes of lightning _A_ and _B_
+would reach him simultaneously, _i.e._ they would meet just where he is
+situated. Now in reality (considered with reference to the railway
+embankment) he is hastening towards the beam of light coming from _B_,
+whilst he is riding on ahead of the beam of light coming from _A_.
+Hence the observer will see the beam of light emitted from _B_ earlier
+than he will see that emitted from _A_. Observers who take the railway
+train as their reference-body must therefore come to the conclusion
+that the lightning flash _B_ took place earlier than the lightning
+flash _A_. We thus arrive at the important result:
+
+Events which are simultaneous with reference to the embankment are not
+simultaneous with respect to the train, and _vice versa_ (relativity of
+simultaneity). Every reference-body (co-ordinate system) has its own
+particular time; unless we are told the reference-body to which the
+statement of time refers, there is no meaning in a statement of the
+time of an event.
+
+Now before the advent of the theory of relativity it had always tacitly
+been assumed in physics that the statement of time had an absolute
+significance, _i.e._ that it is independent of the state of motion of
+the body of reference. But we have just seen that this assumption is
+incompatible with the most natural definition of simultaneity; if we
+discard this assumption, then the conflict between the law of the
+propagation of light _in vacuo_ and the principle of relativity
+(developed in Section VII) disappears.
+
+We were led to that conflict by the considerations of Section VI, which
+are now no longer tenable. In that section we concluded that the man in
+the carriage, who traverses the distance _w per second_ relative to the
+carriage, traverses the same distance also with respect to the
+embankment _in each second_ of time. But, according to the foregoing
+considerations, the time required by a particular occurrence with
+respect to the carriage must not be considered equal to the duration of
+the same occurrence as judged from the embankment (as reference-body).
+Hence it cannot be contended that the man in walking travels the
+distance _w_ relative to the railway line in a time which is equal to
+one second as judged from the embankment.
+
+Moreover, the considerations of Section VI are based on yet a second
+assumption, which, in the light of a strict consideration, appears to
+be arbitrary, although it was always tacitly made even before the
+introduction of the theory of relativity.
+
+
+X.
+
+ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
+
+Let us consider two particular points on the train [8] travelling along
+the embankment with the velocity _v_, and inquire as to their distance
+apart. We already know that it is necessary to have a body of reference
+for the measurement of a distance, with respect to which body the
+distance can be measured up. It is the simplest plan to use the train
+itself as reference-body (co-ordinate system). An observer in the train
+measures the interval by marking off his measuring-rod in a straight
+line (_e.g._ along the floor of the carriage) as many times as is
+necessary to take him from the one marked point to the other. Then the
+number which tells us how often the rod has to be laid down is the
+required distance.
+
+
+ [8] _e.g._ the middle of the first and of the hundredth carriage.
+
+
+It is a different matter when the distance has to be judged from the
+railway line. Here the following method suggests itself. If we call
+_A′_ and _B′_ the two points on the train whose distance apart is
+required, then both of these points are moving with the velocity v
+along the embankment. In the first place we require to determine the
+points _A_ and _B_ of the embankment which are just being passed by the
+two points _A′_ and _B′_ at a particular time t—judged from the
+embankment. These points _A_ and _B_ of the embankment can be
+determined by applying the definition of time given in Section VIII.
+The distance between these points A and B is then measured by repeated
+application of the measuring-rod along the embankment.
+
+_A priori_ it is by no means certain that this last measurement will
+supply us with the same result as the first. Thus the length of the
+train as measured from the embankment may be different from that
+obtained by measuring in the train itself. This circumstance leads us
+to a second objection which must be raised against the apparently
+obvious consideration of Section VI. Namely, if the man in the carriage
+covers the distance _w_ in a unit of time—_measured from the
+train_,—then this distance—_as measured from the embankment_ is not
+necessarily also equal to _w_.
+
+
+XI.
+
+THE LORENTZ TRANSFORMATION
+
+The results of the last three sections show that the apparent
+incompatibility of the law of propagation of light with the principle
+of relativity (Section VII) has been derived by means of a
+consideration which borrowed two unjustifiable hypotheses from
+classical mechanics; these are as follows:
+
+(1) The time-interval (time) between two events is independent of the
+condition of motion of the body of reference.
+
+
+(2) The space-interval (distance) between two points of a rigid body is
+independent of the condition of motion of the body of reference.
+
+
+If we drop these hypotheses, then the dilemma of Section VII
+disappears, because the theorem of the addition of velocities derived
+in Section VI becomes invalid. The possibility presents itself that the
+law of the propagation of light _in vacuo_ may be compatible with the
+principle of relativity, and the question arises: How have we to modify
+the considerations of Section VI in order to remove the apparent
+disagreement between these two fundamental results of experience? This
+question leads to a general one. In the discussion of Section VI we
+have to do with places and times relative both to the train and to the
+embankment. How are we to find the place and time of an event in
+relation to the train, when we know the place and time of the event
+with respect to the railway embankment? Is there a thinkable answer to
+this question of such a nature that the law of transmission of light
+_in vacuo_ does not contradict the principle of relativity? In other
+words: Can we conceive of a relation between place and time of the
+individual events relative to both reference-bodies, such that every
+ray of light possesses the velocity of transmission _c_ relative to the
+embankment and relative to the train? This question leads to a quite
+definite positive answer, and to a perfectly definite transformation
+law for the space-time magnitudes of an event when changing over from
+one body of reference to another.
+
+Before we deal with this, we shall introduce the following incidental
+consideration. Up to the present we have only considered events taking
+place along the embankment, which had mathematically to assume the
+function of a straight line. In the manner indicated in Section II we
+can imagine this reference-body supplemented laterally and in a
+vertical direction by means of a framework of rods, so that an event
+which takes place anywhere can be localised with reference to this
+framework. Similarly, we can imagine the train travelling with the
+velocity _v_ to be continued across the whole of space, so that every
+event, no matter how far off it may be, could also be localised with
+respect to the second framework. Without committing any fundamental
+error, we can disregard the fact that in reality these frameworks would
+continually interfere with each other, owing to the impenetrability of
+solid bodies. In every such framework we imagine three surfaces
+perpendicular to each other marked out, and designated as “co-ordinate
+planes” (“co-ordinate system”). A co-ordinate system _K_ then
+corresponds to the embankment, and a co-ordinate system _K′_ to the
+train. An event, wherever it may have taken place, would be fixed in
+space with respect to _K_ by the three perpendiculars _x, y, z_ on the
+co-ordinate planes, and with regard to time by a time value _t_.
+Relative to _K′, the same event_ would be fixed in respect of space and
+time by corresponding values _x′, y′, z′, t′_, which of course are not
+identical with _x, y, z, t_. It has already been set forth in detail
+how these magnitudes are to be regarded as results of physical
+measurements.
+
+image002
+
+
+Obviously our problem can be exactly formulated in the following
+manner. What are the values _x′, y′, z′, t′_, of an event with respect
+to _K′_, when the magnitudes _x, y, z, t_, of the same event with
+respect to _K_ are given? The relations must be so chosen that the law
+of the transmission of light in vacuo is satisfied for one and the same
+ray of light (and of course for every ray) with respect to _K_ and
+_K′_. For the relative orientation in space of the co-ordinate systems
+indicated in the diagram (Fig. 2), this problem is solved by means of
+the equations:
+
+image003
+
+
+_y′_ = _y_
+
+_z′_ = _z_
+
+
+image004
+
+
+This system of equations is known as the “Lorentz transformation.”[9]
+
+
+ [9] A simple derivation of the Lorentz transformation is given in
+ Appendix I.
+
+
+If in place of the law of transmission of light we had taken as our
+basis the tacit assumptions of the older mechanics as to the absolute
+character of times and lengths, then instead of the above we should
+have obtained the following equations:
+
+_x′_ = _x_ – _vt_
+
+
+_y′_ = _y_
+
+
+_z′_ = _z_
+
+
+_t′_ = _t_
+
+
+This system of equations is often termed the “Galilei transformation.”
+The Galilei transformation can be obtained from the Lorentz
+transformation by substituting an infinitely large value for the
+velocity of light _c_ in the latter transformation.
+
+Aided by the following illustration, we can readily see that, in
+accordance with the Lorentz transformation, the law of the transmission
+of light _in vacuo_ is satisfied both for the reference-body _K_ and
+for the reference-body _K′_. A light-signal is sent along the positive
+_x_-axis, and this light-stimulus advances in accordance with the
+equation
+
+_x_ = _ct_,
+
+
+_i.e._ with the velocity _c_. According to the equations of the Lorentz
+transformation, this simple relation between _x_ and _t_ involves a
+relation between _x′_ and _t′_. In point of fact, if we substitute for
+_x_ the value _ct_ in the first and fourth equations of the Lorentz
+transformation, we obtain:
+
+image005
+
+
+from which, by division, the expression
+
+_x′_ = _ct′_
+
+
+immediately follows. If referred to the system _K′_, the propagation of
+light takes place according to this equation. We thus see that the
+velocity of transmission relative to the reference-body _K′_ is also
+equal to _c_. The same result is obtained for rays of light advancing
+in any other direction whatsoever. Of cause this is not surprising,
+since the equations of the Lorentz transformation were derived
+conformably to this point of view.
+
+
+XII.
+
+THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
+
+Place a metre-rod in the _x′_-axis of _K′_ in such a manner that one
+end (the beginning) coincides with the point _x′_ = 0 whilst the other
+end (the end of the rod) coincides with the point _x′_ = 1. What is the
+length of the metre-rod relatively to the system _K_? In order to learn
+this, we need only ask where the beginning of the rod and the end of
+the rod lie with respect to _K_ at a particular time _t_ of the system
+_K_. By means of the first equation of the Lorentz transformation the
+values of these two points at the time _t_ = 0 can be shown to be
+
+image006
+
+
+the distance between the points being
+
+image007
+
+
+But the metre-rod is moving with the velocity _v_ relative to _K_. It
+therefore follows that the length of a rigid metre-rod moving in the
+direction of its length with a velocity _v_ is
+
+image008
+
+
+of a metre. The rigid rod is thus shorter when in motion than when at
+rest, and the more quickly it is moving, the shorter is the rod. For
+the velocity _v_ = _c_ we should have
+
+image009
+
+
+and for still greater velocities the square-root becomes imaginary.
+From this we conclude that in the theory of relativity the velocity _c_
+plays the part of a limiting velocity, which can neither be reached nor
+exceeded by any real body.
+
+Of course this feature of the velocity _c_ as a limiting velocity also
+clearly follows from the equations of the Lorentz transformation, for
+these became meaningless if we choose values of _v_ greater than _c_.
+
+If, on the contrary, we had considered a metre-rod at rest in the
+_x_-axis with respect to _K_, then we should have found that the length
+of the rod as judged from _K′_ would have been
+
+image010
+
+
+this is quite in accordance with the principle of relativity which
+forms the basis of our considerations.
+
+_A priori_ it is quite clear that we must be able to learn something
+about the physical behaviour of measuring-rods and clocks from the
+equations of transformation, for the magnitudes _z, y, x, t_, are
+nothing more nor less than the results of measurements obtainable by
+means of measuring-rods and clocks. If we had based our considerations
+on the Galileian transformation we should not have obtained a
+contraction of the rod as a consequence of its motion.
+
+Let us now consider a seconds-clock which is permanently situated at
+the origin (_x′_ = 0) of _K′_. _t′_ = 0 and _t′_ = 1 are two successive
+ticks of this clock. The first and fourth equations of the Lorentz
+transformation give for these two ticks:
+
+_t_ = 0
+
+and
+
+image011
+
+
+As judged from _K_, the clock is moving with the velocity _v_; as
+judged from this reference-body, the time which elapses between two
+strokes of the clock is not one second, but
+
+image012
+
+
+seconds, _i.e._ a somewhat larger time. As a consequence of its motion
+the clock goes more slowly than when at rest. Here also the velocity
+_c_ plays the part of an unattainable limiting velocity.
+
+
+XIII.
+
+THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU
+
+Now in practice we can move clocks and measuring-rods only with
+velocities that are small compared with the velocity of light; hence we
+shall hardly be able to compare the results of the previous section
+directly with the reality. But, on the other hand, these results must
+strike you as being very singular, and for that reason I shall now draw
+another conclusion from the theory, one which can easily be derived
+from the foregoing considerations, and which has been most elegantly
+confirmed by experiment.
+
+In Section VI we derived the theorem of the addition of velocities in
+one direction in the form which also results from the hypotheses of
+classical mechanics. This theorem can also be deduced readily from the
+Galilei transformation (Section XI). In place of the man walking inside
+the carriage, we introduce a point moving relatively to the co-ordinate
+system _K′_ in accordance with the equation
+
+_x′_ = _wt′_
+
+By means of the first and fourth equations of the Galilei
+transformation we can express _x′_ and _t′_ in terms of _x_ and _t_,
+and we then obtain
+
+_x_ = (_v_ + _w_)_t_
+
+This equation expresses nothing else than the law of motion of the
+point with reference to the system _K_ (of the man with reference to
+the embankment). We denote this velocity by the symbol _W_, and we then
+obtain, as in Section VI,
+
+_W_ = _v_ + _w_ . . . . . . . (A).
+
+But we can carry out this consideration just as well on the basis of
+the theory of relativity. In the equation
+
+_x′_ = _wt′_
+
+we must then express _x′_ and _t′_ in terms of _x_ and _t_, making use
+of the first and fourth equations of the _Lorentz transformation_.
+Instead of the equation (A) we then obtain the equation
+
+image013
+
+
+which corresponds to the theorem of addition for velocities in one
+direction according to the theory of relativity. The question now
+arises as to which of these two theorems is the better in accord with
+experience. On this point we are enlightened by a most important
+experiment which the brilliant physicist Fizeau performed more than
+half a century ago, and which has been repeated since then by some of
+the best experimental physicists, so that there can be no doubt about
+its result. The experiment is concerned with the following question.
+Light travels in a motionless liquid with a particular velocity _w_.
+How quickly does it travel in the direction of the arrow in the tube
+_T_ (see the accompanying diagram, Fig. 3) when the liquid above
+mentioned is flowing through the tube with a velocity _v_?
+
+image014
+
+
+In accordance with the principle of relativity we shall certainly have
+to take for granted that the propagation of light always takes place
+with the same velocity _w with respect to the liquid_, whether the
+latter is in motion with reference to other bodies or not. The velocity
+of light relative to the liquid and the velocity of the latter relative
+to the tube are thus known, and we require the velocity of light
+relative to the tube.
+
+It is clear that we have the problem of Section VI again before us. The
+tube plays the part of the railway embankment or of the co-ordinate
+system _K_, the liquid plays the part of the carriage or of the
+co-ordinate system _K′_, and finally, the light plays the part of the
+man walking along the carriage, or of the moving point in the present
+section. If we denote the velocity of the light relative to the tube by
+_W_, then this is given by the equation (A) or (B), according as the
+Galilei transformation or the Lorentz transformation corresponds to the
+facts. Experiment[10] decides in favour of equation (B) derived from
+the theory of relativity, and the agreement is, indeed, very exact.
+According to recent and most excellent measurements by Zeeman, the
+influence of the velocity of flow _v_ on the propagation of light is
+represented by formula (B) to within one per cent.
+
+
+ [10] Fizeau found
+
+
+image015
+
+
+where
+
+
+image016
+
+
+is the index of refraction of the liquid. On the other hand, owing to
+the smallness of
+
+
+image017
+
+
+as compared with 1, we can replace (B) in the first place by
+
+
+image018
+
+
+or to the same order of approximation by
+
+
+image019
+
+
+which agrees with Fizeau’s result.
+
+
+Nevertheless we must now draw attention to the fact that a theory of
+this phenomenon was given by H. A. Lorentz long before the statement of
+the theory of relativity. This theory was of a purely electrodynamical
+nature, and was obtained by the use of particular hypotheses as to the
+electromagnetic structure of matter. This circumstance, however, does
+not in the least diminish the conclusiveness of the experiment as a
+crucial test in favour of the theory of relativity, for the
+electrodynamics of Maxwell-Lorentz, on which the original theory was
+based, in no way opposes the theory of relativity. Rather has the
+latter been developed trom electrodynamics as an astoundingly simple
+combination and generalisation of the hypotheses, formerly independent
+of each other, on which electrodynamics was built.
+
+
+XIV.
+
+THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
+
+Our train of thought in the foregoing pages can be epitomised in the
+following manner. Experience has led to the conviction that, on the one
+hand, the principle of relativity holds true and that on the other hand
+the velocity of transmission of light _in vacuo_ has to be considered
+equal to a constant _c_. By uniting these two postulates we obtained
+the law of transformation for the rectangular co-ordinates _x, y, z_
+and the time _t_ of the events which constitute the processes of
+nature. In this connection we did not obtain the Galilei
+transformation, but, differing from classical mechanics, the _Lorentz
+transformation_.
+
+The law of transmission of light, the acceptance of which is justified
+by our actual knowledge, played an important part in this process of
+thought. Once in possession of the Lorentz transformation, however, we
+can combine this with the principle of relativity, and sum up the
+theory thus:
+
+Every general law of nature must be so constituted that it is
+transformed into a law of exactly the same form when, instead of the
+space-time variables _x, y, z, t_ of the original coordinate system
+_K_, we introduce new space-time variables _x′, y′, z′, t′_ of a
+co-ordinate system _K′_. In this connection the relation between the
+ordinary and the accented magnitudes is given by the Lorentz
+transformation. Or in brief: General laws of nature are co-variant with
+respect to Lorentz transformations.
+
+This is a definite mathematical condition that the theory of relativity
+demands of a natural law, and in virtue of this, the theory becomes a
+valuable heuristic aid in the search for general laws of nature. If a
+general law of nature were to be found which did not satisfy this
+condition, then at least one of the two fundamental assumptions of the
+theory would have been disproved. Let us now examine what general
+results the latter theory has hitherto evinced.
+
+
+XV.
+
+GENERAL RESULTS OF THE THEORY
+
+It is clear from our previous considerations that the (special) theory
+of relativity has grown out of electrodynamics and optics. In these
+fields it has not appreciably altered the predictions of theory, but it
+has considerably simplified the theoretical structure, _i.e._ the
+derivation of laws, and—what is incomparably more important—it has
+considerably reduced the number of independent hypotheses forming the
+basis of theory. The special theory of relativity has rendered the
+Maxwell-Lorentz theory so plausible, that the latter would have been
+generally accepted by physicists even if experiment had decided less
+unequivocally in its favour.
+
+Classical mechanics required to be modified before it could come into
+line with the demands of the special theory of relativity. For the main
+part, however, this modification affects only the laws for rapid
+motions, in which the velocities of matter _v_ are not very small as
+compared with the velocity of light. We have experience of such rapid
+motions only in the case of electrons and ions; for other motions the
+variations from the laws of classical mechanics are too small to make
+themselves evident in practice. We shall not consider the motion of
+stars until we come to speak of the general theory of relativity. In
+accordance with the theory of relativity the kinetic energy of a
+material point of mass _m_ is no longer given by the well-known
+expression
+
+image020
+
+
+but by the expression
+
+image021
+
+
+This expression approaches infinity as the velocity _v_ approaches the
+velocity of light _c_. The velocity must therefore always remain less
+than _c_, however great may be the energies used to produce the
+acceleration. If we develop the expression for the kinetic energy in
+the form of a series, we obtain
+
+image022
+
+
+When
+
+image023
+
+
+is small compared with unity, the third of these terms is always small
+in comparison with the second, which last is alone considered in
+classical mechanics. The first term _mc_2 does not contain the
+velocity, and requires no consideration if we are only dealing with the
+question as to how the energy of a point-mass; depends on the velocity.
+We shall speak of its essential significance later.
+
+The most important result of a general character to which the special
+theory of relativity has led is concerned with the conception of mass.
+Before the advent of relativity, physics recognised two conservation
+laws of fundamental importance, namely, the law of the conservation of
+energy and the law of the conservation of mass these two fundamental
+laws appeared to be quite independent of each other. By means of the
+theory of relativity they have been united into one law. We shall now
+briefly consider how this unification came about, and what meaning is
+to be attached to it.
+
+The principle of relativity requires that the law of the conservation
+of energy should hold not only with reference to a co-ordinate system
+_K_, but also with respect to every co-ordinate system _K′_ which is in
+a state of uniform motion of translation relative to _K_, or, briefly,
+relative to every “Galileian” system of co-ordinates. In contrast to
+classical mechanics; the Lorentz transformation is the deciding factor
+in the transition from one such system to another.
+
+By means of comparatively simple considerations we are led to draw the
+following conclusion from these premises, in conjunction with the
+fundamental equations of the electrodynamics of Maxwell: A body moving
+with the velocity _v_, which absorbs[11] an amount of energy _E_0 in
+the form of radiation without suffering an alteration in velocity in
+the process, has, as a consequence, its energy increased by an amount
+
+image024
+
+
+
+ [11] _E_0 is the energy taken up, as judged from a co-ordinate system
+ moving with the body.
+
+
+In consideration of the expression given above for the kinetic energy
+of the body, the required energy of the body comes out to be
+
+image025
+
+
+Thus the body has the same energy as a body of mass
+
+image026
+
+
+moving with the velocity _v_. Hence we can say: If a body takes up an
+amount of energy _E_0, then its inertial mass increases by an amount
+
+image027
+
+
+the inertial mass of a body is not a constant but varies according to
+the change in the energy of the body. The inertial mass of a system of
+bodies can even be regarded as a measure of its energy. The law of the
+conservation of the mass of a system becomes identical with the law of
+the conservation of energy, and is only valid provided that the system
+neither takes up nor sends out energy. Writing the expression for the
+energy in the form
+
+image028
+
+
+we see that the term _mc_2, which has hitherto attracted our attention,
+is nothing else than the energy possessed by the body[12] before it
+absorbed the energy _E_0.
+
+
+ [12] As judged from a co-ordinate system moving with the body.
+
+
+A direct comparison of this relation with experiment is not possible at
+the present time (1920; see[Note], p. 48), owing to the fact that the
+changes in energy _E_0 to which we can subject a system are not large
+enough to make themselves perceptible as a change in the inertial mass
+of the system.
+
+image027
+
+
+is too small in comparison with the mass _m_, which was present before
+the alteration of the energy. It is owing to this circumstance that
+classical mechanics was able to establish successfully the conservation
+of mass as a law of independent validity.
+
+
+ [Note] The equation E = mc2 has been thoroughly proved time and again
+ since this time.
+
+
+Let me add a final remark of a fundamental nature. The success of the
+Faraday-Maxwell interpretation of electromagnetic action at a distance
+resulted in physicists becoming convinced that there are no such things
+as instantaneous actions at a distance (not involving an intermediary
+medium) of the type of Newton’s law of gravitation.
+
+According to the theory of relativity, action at a distance with the
+velocity of light always takes the place of instantaneous action at a
+distance or of action at a distance with an infinite velocity of
+transmission. This is connected with the fact that the velocity _c_
+plays a fundamental role in this theory. In Part II we shall see in
+what way this result becomes modified in the general theory of
+relativity.
+
+
+XVI.
+
+EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
+
+To what extent is the special theory of relativity supported by
+experience? This question is not easily answered for the reason already
+mentioned in connection with the fundamental experiment of Fizeau. The
+special theory of relativity has crystallised out from the
+Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
+experience which support the electromagnetic theory also support the
+theory of relativity. As being of particular importance, I mention here
+the fact that the theory of relativity enables us to predict the
+effects produced on the light reaching us from the fixed stars. These
+results are obtained in an exceedingly simple manner, and the effects
+indicated, which are due to the relative motion of the earth with
+reference to those fixed stars are found to be in accord with
+experience. We refer to the yearly movement of the apparent position of
+the fixed stars resulting from the motion of the earth round the sun
+(aberration), and to the influence of the radial components of the
+relative motions of the fixed stars with respect to the earth on the
+colour of the light reaching us from them. The latter effect manifests
+itself in a slight displacement of the spectral lines of the light
+transmitted to us from a fixed star, as compared with the position of
+the same spectral lines when they are produced by a terrestrial source
+of light (Doppler principle). The experimental arguments in favour of
+the Maxwell-Lorentz theory, which are at the same time arguments in
+favour of the theory of relativity, are too numerous to be set forth
+here. In reality they limit the theoretical possibilities to such an
+extent, that no other theory than that of Maxwell and Lorentz has been
+able to hold its own when tested by experience.
+
+But there are two classes of experimental facts hitherto obtained which
+can be represented in the Maxwell-Lorentz theory only by the
+introduction of an auxiliary hypothesis, which in itself—_i.e._ without
+making use of the theory of relativity—appears extraneous.
+
+It is known that cathode rays and the so-called β-rays emitted by
+radioactive substances consist of negatively electrified particles
+(electrons) of very small inertia and large velocity. By examining the
+deflection of these rays under the influence of electric and magnetic
+fields, we can study the law of motion of these particles very exactly.
+
+In the theoretical treatment of these electrons, we are faced with the
+difficulty that electrodynamic theory of itself is unable to give an
+account of their nature. For since electrical masses of one sign repel
+each other, the negative electrical masses constituting the electron
+would necessarily be scattered under the influence of their mutual
+repulsions, unless there are forces of another kind operating between
+them, the nature of which has hitherto remained obscure to us.[13] If
+we now assume that the relative distances between the electrical masses
+constituting the electron remain unchanged during the motion of the
+electron (rigid connection in the sense of classical mechanics), we
+arrive at a law of motion of the electron which does not agree with
+experience. Guided by purely formal points of view, H. A. Lorentz was
+the first to introduce the hypothesis that the form of the electron
+experiences a contraction in the direction of motion in consequence of
+that motion. the contracted length being proportional to the expression
+
+image029
+
+
+This, hypothesis, which is not justifiable by any electrodynamical
+facts, supplies us then with that particular law of motion which has
+been confirmed with great precision in recent years.
+
+
+ [13] The general theory of relativity renders it likely that the
+ electrical masses of an electron are held together by gravitational
+ forces.
+
+
+The theory of relativity leads to the same law of motion, without
+requiring any special hypothesis whatsoever as to the structure and the
+behaviour of the electron. We arrived at a similar conclusion in
+Section XIII in connection with the experiment of Fizeau, the result of
+which is foretold by the theory of relativity without the necessity of
+drawing on hypotheses as to the physical nature of the liquid.
+
+The second class of facts to which we have alluded has reference to the
+question whether or not the motion of the earth in space can be made
+perceptible in terrestrial experiments. We have already remarked in
+Section V that all attempts of this nature led to a negative result.
+Before the theory of relativity was put forward, it was difficult to
+become reconciled to this negative result, for reasons now to be
+discussed. The inherited prejudices about time and space did not allow
+any doubt to arise as to the prime importance of the Galileian
+transformation for changing over from one body of reference to another.
+Now assuming that the Maxwell-Lorentz equations hold for a
+reference-body _K_, we then find that they do not hold for a
+reference-body _K′_ moving uniformly with respect to _K_, if we assume
+that the relations of the Galileian transformation exist between the
+co-ordinates of _K_ and _K′_. It thus appears that, of all Galileian
+co-ordinate systems, one (_K_) corresponding to a particular state of
+motion is physically unique. This result was interpreted physically by
+regarding _K_ as at rest with respect to a hypothetical æther of space.
+On the other hand, all coordinate systems _K′_ moving relatively to _K_
+were to be regarded as in motion with respect to the æther. To this
+motion of _K′_ against the æther (“æther-drift” relative to _K′_) were
+attributed the more complicated laws which were supposed to hold
+relative to _K′_. Strictly speaking, such an æther-drift ought also to
+be assumed relative to the earth, and for a long time the efforts of
+physicists were devoted to attempts to detect the existence of an
+æther-drift at the earth’s surface.
+
+In one of the most notable of these attempts Michelson devised a method
+which appears as though it must be decisive. Imagine two mirrors so
+arranged on a rigid body that the reflecting surfaces face each other.
+A ray of light requires a perfectly definite time _T_ to pass from one
+mirror to the other and back again, if the whole system be at rest with
+respect to the æther. It is found by calculation, however, that a
+slightly different time _T′_ is required for this process, if the body,
+together with the mirrors, be moving relatively to the æther. And yet
+another point: it is shown by calculation that for a given velocity _v_
+with reference to the æther, this time _T′_ is different when the body
+is moving perpendicularly to the planes of the mirrors from that
+resulting when the motion is parallel to these planes. Although the
+estimated difference between these two times is exceedingly small,
+Michelson and Morley performed an experiment involving interference in
+which this difference should have been clearly detectable. But the
+experiment gave a negative result—a fact very perplexing to physicists.
+Lorentz and FitzGerald rescued the theory from this difficulty by
+assuming that the motion of the body relative to the æther produces a
+contraction of the body in the direction of motion, the amount of
+contraction being just sufficient to compensate for the difference in
+time mentioned above. Comparison with the discussion in Section XII
+shows that also from the standpoint of the theory of relativity this
+solution of the difficulty was the right one. But on the basis of the
+theory of relativity the method of interpretation is incomparably more
+satisfactory. According to this theory there is no such thing as a
+“specially favoured” (unique) co-ordinate system to occasion the
+introduction of the æther-idea, and hence there can be no æther-drift,
+nor any experiment with which to demonstrate it. Here the contraction
+of moving bodies follows from the two fundamental principles of the
+theory, without the introduction of particular hypotheses; and as the
+prime factor involved in this contraction we find, not the motion in
+itself, to which we cannot attach any meaning, but the motion with
+respect to the body of reference chosen in the particular case in
+point. Thus for a co-ordinate system moving with the earth the mirror
+system of Michelson and Morley is not shortened, but it _is_ shortened
+for a co-ordinate system which is at rest relatively to the sun.
+
+
+XVII.
+
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE
+
+The non-mathematician is seized by a mysterious shuddering when he
+hears of “four-dimensional” things, by a feeling not unlike that
+awakened by thoughts of the occult. And yet there is no more
+common-place statement than that the world in which we live is a
+four-dimensional space-time continuum.
+
+Space is a three-dimensional continuum. By this we mean that it is
+possible to describe the position of a point (at rest) by means of
+three numbers (co-ordinates) _x, y, z_, and that there is an indefinite
+number of points in the neighbourhood of this one, the position of
+which can be described by co-ordinates such as _x1, y1, z1_, which may
+be as near as we choose to the respective values of the co-ordinates
+_x, y, z_, of the first point. In virtue of the latter property we
+speak of a “continuum,” and owing to the fact that there are three
+co-ordinates we speak of it as being “three-dimensional.”
+
+Similarly, the world of physical phenomena which was briefly called
+“world” by Minkowski is naturally four dimensional in the space-time
+sense. For it is composed of individual events, each of which is
+described by four numbers, namely, three space co-ordinates _x, y, z_,
+and a time co-ordinate, the time value _t_. The “world” is in this
+sense also a continuum; for to every event there are as many
+“neighbouring” events (realised or at least thinkable) as we care to
+choose, the co-ordinates _x1, y1, z1, t1_ of which differ by an
+indefinitely small amount from those of the event _x, y, z, t_
+originally considered. That we have not been accustomed to regard the
+world in this sense as a four-dimensional continuum is due to the fact
+that in physics, before the advent of the theory of relativity, time
+played a different and more independent rôle, as compared with the
+space coordinates. It is for this reason that we have been in the habit
+of treating time as an independent continuum. As a matter of fact,
+according to classical mechanics, time is absolute, _i.e._ it is
+independent of the position and the condition of motion of the system
+of co-ordinates. We see this expressed in the last equation of the
+Galileian transformation (_t′_ = _t_).
+
+The four-dimensional mode of consideration of the “world” is natural on
+the theory of relativity, since according to this theory time is robbed
+of its independence. This is shown by the fourth equation of the
+Lorentz transformation:
+
+image030
+
+
+Moreover, according to this equation the time difference Δ_t′_ of two
+events with respect to _K′_ does not in general vanish, even when the
+time difference Δ_t_ of the same events with reference to _K_ vanishes.
+Pure “space-distance” of two events with respect to _K_ results in
+“time-distance ” of the same events with respect to _K_. But the
+discovery of Minkowski, which was of importance for the formal
+development of the theory of relativity, does not lie here. It is to be
+found rather in the fact of his recognition that the four-dimensional
+space-time continuum of the theory of relativity, in its most essential
+formal properties, shows a pronounced relationship to the
+three-dimensional continuum of Euclidean geometrical space.[14] In
+order to give due prominence to this relationship, however, we must
+replace the usual time co-ordinate t by an imaginary magnitude
+
+image031
+
+
+proportional to it. Under these conditions, the natural laws satisfying
+the demands of the (special) theory of relativity assume mathematical
+forms, in which the time co-ordinate plays exactly the same role as the
+three space co-ordinates. Formally, these four co-ordinates correspond
+exactly to the three space co-ordinates in Euclidean geometry. It must
+be clear even to the non-mathematician that, as a consequence of this
+purely formal addition to our knowledge, the theory perforce gained
+clearness in no mean measure.
+
+
+ [14] Cf. the somewhat more detailed discussion in Appendix II.
+
+
+These inadequate remarks can give the reader only a vague notion of the
+important idea contributed by Minkowski. Without it the general theory
+of relativity, of which the fundamental ideas are developed in the
+following pages, would perhaps have got no farther than its long
+clothes. Minkowski’s work is doubtless difficult of access to anyone
+inexperienced in mathematics, but since it is not necessary to have a
+very exact grasp of this work in order to understand the fundamental
+ideas of either the special or the general theory of relativity, I
+shall leave it here at present, and revert to it only towards the end
+of Part II.
+
+
+PART II: THE GENERAL THEORY OF RELATIVITY
+
+
+XVIII.
+
+SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
+
+The basal principle, which was the pivot of all our previous
+considerations, was the _special_ principle of relativity, _i.e._ the
+principle of the physical relativity of all _uniform_ motion. Let as
+once more analyse its meaning carefully.
+
+It was at all times clear that, from the point of view of the idea it
+conveys to us, every motion must be considered only as a relative
+motion. Returning to the illustration we have frequently used of the
+embankment and the railway carriage, we can express the fact of the
+motion here taking place in the following two forms, both of which are
+equally justifiable:
+
+(_a_) The carriage is in motion relative to the embankment,
+
+
+(_b_) The embankment is in motion relative to the carriage.
+
+
+In (_a_) the embankment, in (_b_) the carriage, serves as the body of
+reference in our statement of the motion taking place. If it is simply
+a question of detecting or of describing the motion involved, it is in
+principle immaterial to what reference-body we refer the motion. As
+already mentioned, this is self-evident, but it must not be confused
+with the much more comprehensive statement called “the principle of
+relativity,” which we have taken as the basis of our investigations.
+
+The principle we have made use of not only maintains that we may
+equally well choose the carriage or the embankment as our
+reference-body for the description of any event (for this, too, is
+self-evident). Our principle rather asserts what follows: If we
+formulate the general laws of nature as they are obtained from
+experience, by making use of
+
+(_a_) the embankment as reference-body,
+
+
+(_b_) the railway carriage as reference-body,
+
+
+then these general laws of nature (_e.g._ the laws of mechanics or the
+law of the propagation of light _in vacuo_) have exactly the same form
+in both cases. This can also be expressed as follows: For the physical
+description of natural processes, neither of the reference bodies _K,
+K′_ is unique (lit. “specially marked out”) as compared with the other.
+Unlike the first, this latter statement need not of necessity hold _a
+priori;_ it is not contained in the conceptions of “motion” and
+“reference-body” and derivable from them; only _experience_ can decide
+as to its correctness or incorrectness.
+
+Up to the present, however, we have by no means maintained the
+equivalence of _all_ bodies of reference _K_ in connection with the
+formulation of natural laws. Our course was more on the following
+Iines. In the first place, we started out from the assumption that
+there exists a reference-body _K_, whose condition of motion is such
+that the Galileian law holds with respect to it: A particle left to
+itself and sufficiently far removed from all other particles moves
+uniformly in a straight line. With reference to K (Galileian
+reference-body) the laws of nature were to be as simple as possible.
+But in addition to K, all bodies of reference _K′_ should be given
+preference in this sense, and they should be exactly equivalent to _K_
+for the formulation of natural laws, provided that they are in a state
+of _uniform rectilinear and non-rotary motion_ with respect to _K_; all
+these bodies of reference are to be regarded as Galileian
+reference-bodies. The validity of the principle of relativity was
+assumed only for these reference-bodies, but not for others (_e.g._
+those possessing motion of a different kind). In this sense we speak of
+the _special_ principle of relativity, or special theory of relativity.
+
+In contrast to this we wish to understand by the “general principle of
+relativity” the following statement: All bodies of reference _K, K′_,
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion. But before proceeding farther, it ought to be pointed
+out that this formulation must be replaced later by a more abstract
+one, for reasons which will become evident at a later stage.
+
+Since the introduction of the special principle of relativity has been
+justified, every intellect which strives after generalisation must feel
+the temptation to venture the step towards the general principle of
+relativity. But a simple and apparently quite reliable consideration
+seems to suggest that, for the present at any rate, there is little
+hope of success in such an attempt; Let us imagine ourselves
+transferred to our old friend the railway carriage, which is travelling
+at a uniform rate. As long as it is moving uniformly, the occupant of
+the carriage is not sensible of its motion, and it is for this reason
+that he can without reluctance interpret the facts of the case as
+indicating that the carriage is at rest, but the embankment in motion.
+Moreover, according to the special principle of relativity, this
+interpretation is quite justified also from a physical point of view.
+If the motion of the carriage is now changed into a non-uniform motion,
+as for instance by a powerful application of the brakes, then the
+occupant of the carriage experiences a correspondingly powerful jerk
+forwards. The retarded motion is manifested in the mechanical behaviour
+of bodies relative to the person in the railway carriage. The
+mechanical behaviour is different from that of the case previously
+considered, and for this reason it would appear to be impossible that
+the same mechanical laws hold relatively to the non-uniformly moving
+carriage, as hold with reference to the carriage when at rest or in
+uniform motion. At all events it is clear that the Galileian law does
+not hold with respect to the non-uniformly moving carriage. Because of
+this, we feel compelled at the present juncture to grant a kind of
+absolute physical reality to non-uniform motion, in opposition to the
+general principle of relativity. But in what follows we shall soon see
+that this conclusion cannot be maintained.
+
+
+XIX.
+
+THE GRAVITATIONAL FIELD
+
+“If we pick up a stone and then let it go, why does it fall to the
+ground?” The usual answer to this question is: “Because it is attracted
+by the earth.” Modern physics formulates the answer rather differently
+for the following reason. As a result of the more careful study of
+electromagnetic phenomena, we have come to regard action at a distance
+as a process impossible without the intervention of some intermediary
+medium. If, for instance, a magnet attracts a piece of iron, we cannot
+be content to regard this as meaning that the magnet acts directly on
+the iron through the intermediate empty space, but we are constrained
+to imagine—after the manner of Faraday—that the magnet always calls
+into being something physically real in the space around it, that
+something being what we call a “magnetic field.” In its turn this
+magnetic field operates on the piece of iron, so that the latter
+strives to move towards the magnet. We shall not discuss here the
+justification for this incidental conception, which is indeed a
+somewhat arbitrary one. We shall only mention that with its aid
+electromagnetic phenomena can be theoretically represented much more
+satisfactorily than without it, and this applies particularly to the
+transmission of electromagnetic waves. The effects of gravitation also
+are regarded in an analogous manner.
+
+The action of the earth on the stone takes place indirectly. The earth
+produces in its surrounding a gravitational field, which acts on the
+stone and produces its motion of fall. As we know from experience, the
+intensity of the action on a body dimishes according to a quite
+definite law, as we proceed farther and farther away from the earth.
+From our point of view this means: The law governing the properties of
+the gravitational field in space must be a perfectly definite one, in
+order correctly to represent the diminution of gravitational action
+with the distance from operative bodies. It is something like this: The
+body (_e.g._ the earth) produces a field in its immediate neighbourhood
+directly; the intensity and direction of the field at points farther
+removed from the body are thence determined by the law which governs
+the properties in space of the gravitational fields themselves.
+
+In contrast to electric and magnetic fields, the gravitational field
+exhibits a most remarkable property, which is of fundamental importance
+for what follows. Bodies which are moving under the sole influence of a
+gravitational field receive an acceleration, _which does not in the
+least depend either on the material or on the physical state of the
+body._ For instance, a piece of lead and a piece of wood fall in
+exactly the same manner in a gravitational field (_in vacuo_), when
+they start off from rest or with the same initial velocity. This law,
+which holds most accurately, can be expressed in a different form in
+the light of the following consideration.
+
+According to Newton’s law of motion, we have
+
+(Force) = (inertial mass) x (acceleration),
+
+where the “inertial mass” is a characteristic constant of the
+accelerated body. If now gravitation is the cause of the acceleration,
+we then have
+
+(Force) = (gravitational mass) x (intensity of the gravitational
+field),
+
+where the “gravitational mass” is likewise a characteristic constant
+for the body. From these two relations follows:
+
+image032
+
+
+If now, as we find from experience, the acceleration is to be
+independent of the nature and the condition of the body and always the
+same for a given gravitational field, then the ratio of the
+gravitational to the inertial mass must likewise be the same for all
+bodies. By a suitable choice of units we can thus make this ratio equal
+to unity. We then have the following law: The _gravitational_ mass of a
+body is equal to its _inertial_ mass.
+
+It is true that this important law had hitherto been recorded in
+mechanics, but it had not been _interpreted_. A satisfactory
+interpretation can be obtained only if we recognise the following fact:
+_The same_ quality of a body manifests itself according to
+circumstances as “inertia” or as “weight” (lit. “heaviness”). In the
+following section we shall show to what extent this is actually the
+case, and how this question is connected with the general postulate of
+relativity.
+
+
+XX.
+
+THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE
+GENERAL POSTULATE OF RELATIVITY
+
+We imagine a large portion of empty space, so far removed from stars
+and other appreciable masses, that we have before us approximately the
+conditions required by the fundamental law of Galilei. It is then
+possible to choose a Galileian reference-body for this part of space
+(world), relative to which points at rest remain at rest and points in
+motion continue permanently in uniform rectilinear motion. As
+reference-body let us imagine a spacious chest resembling a room with
+an observer inside who is equipped with apparatus. Gravitation
+naturally does not exist for this observer. He must fasten himself with
+strings to the floor, otherwise the slightest impact against the floor
+will cause him to rise slowly towards the ceiling of the room.
+
+To the middle of the lid of the chest is fixed externally a hook with
+rope attached, and now a “being” (what kind of a being is immaterial to
+us) begins pulling at this with a constant force. The chest together
+with the observer then begin to move “upwards” with a uniformly
+accelerated motion. In course of time their velocity will reach
+unheard-of values—provided that we are viewing all this from another
+reference-body which is not being pulled with a rope.
+
+But how does the man in the chest regard the Process? The acceleration
+of the chest will be transmitted to him by the reaction of the floor of
+the chest. He must therefore take up this pressure by means of his legs
+if he does not wish to be laid out full length on the floor. He is then
+standing in the chest in exactly the same way as anyone stands in a
+room of a home on our earth. If he releases a body which he previously
+had in his land, the accelertion of the chest will no longer be
+transmitted to this body, and for this reason the body will approach
+the floor of the chest with an accelerated relative motion. The
+observer will further convince himself _that the acceleration of the
+body towards the floor of the chest is always of the same magnitude,
+whatever kind of body he may happen to use for the experiment._
+
+Relying on his knowledge of the gravitational field (as it was
+discussed in the preceding section), the man in the chest will thus
+come to the conclusion that he and the chest are in a gravitational
+field which is constant with regard to time. Of course he will be
+puzzled for a moment as to why the chest does not fall in this
+gravitational field. just then, however, he discovers the hook in the
+middle of the lid of the chest and the rope which is attached to it,
+and he consequently comes to the conclusion that the chest is suspended
+at rest in the gravitational field.
+
+Ought we to smile at the man and say that he errs in his conclusion? I
+do not believe we ought to if we wish to remain consistent; we must
+rather admit that his mode of grasping the situation violates neither
+reason nor known mechanical laws. Even though it is being accelerated
+with respect to the “Galileian space” first considered, we can
+nevertheless regard the chest as being at rest. We have thus good
+grounds for extending the principle of relativity to include bodies of
+reference which are accelerated with respect to each other, and as a
+result we have gained a powerful argument for a generalised postulate
+of relativity.
+
+We must note carefully that the possibility of this mode of
+interpretation rests on the fundamental property of the gravitational
+field of giving all bodies the same acceleration, or, what comes to the
+same thing, on the law of the equality of inertial and gravitational
+mass. If this natural law did not exist, the man in the accelerated
+chest would not be able to interpret the behaviour of the bodies around
+him on the supposition of a gravitational field, and he would not be
+justified on the grounds of experience in supposing his reference-body
+to be “at rest.”
+
+Suppose that the man in the chest fixes a rope to the inner side of the
+lid, and that he attaches a body to the free end of the rope. The
+result of this will be to stretch the rope so that it will hang
+“vertically” downwards. If we ask for an opinion of the cause of
+tension in the rope, the man in the chest will say: “The suspended body
+experiences a downward force in the gravitational field, and this is
+neutralised by the tension of the rope; what determines the magnitude
+of the tension of the rope is the _gravitational mass_ of the suspended
+body.” On the other hand, an observer who is poised freely in space
+will interpret the condition of things thus: “The rope must perforce
+take part in the accelerated motion of the chest, and it transmits this
+motion to the body attached to it. The tension of the rope is just
+large enough to effect the acceleration of the body. That which
+determines the magnitude of the tension of the rope is the _inertial
+mass_ of the body.” Guided by this example, we see that our extension
+of the principle of relativity implies the _necessity_ of the law of
+the equality of inertial and gravitational mass. Thus we have obtained
+a physical interpretation of this law.
+
+From our consideration of the accelerated chest we see that a general
+theory of relativity must yield important results on the laws of
+gravitation. In point of fact, the systematic pursuit of the general
+idea of relativity has supplied the laws satisfied by the gravitational
+field. Before proceeding farther, however, I must warn the reader
+against a misconception suggested by these considerations. A
+gravitational field exists for the man in the chest, despite the fact
+that there was no such field for the co-ordinate system first chosen.
+Now we might easily suppose that the existence of a gravitational field
+is always only an _apparent_ one. We might also think that, regardless
+of the kind of gravitational field which may be present, we could
+always choose another reference-body such that _no_ gravitational field
+exists with reference to it. This is by no means true for all
+gravitational fields, but only for those of quite special form. It is,
+for instance, impossible to choose a body of reference such that, as
+judged from it, the gravitational field of the earth (in its entirety)
+vanishes.
+
+We can now appreciate why that argument is not convincing, which we
+brought forward against the general principle of relativity at the end
+of Section XVIII. It is certainly true that the observer in the railway
+carriage experiences a jerk forwards as a result of the application of
+the brake, and that he recognises, in this the non-uniformity of motion
+(retardation) of the carriage. But he is compelled by nobody to refer
+this jerk to a “real” acceleration (retardation) of the carriage. He
+might also interpret his experience thus: “My body of reference (the
+carriage) remains permanently at rest. With reference to it, however,
+there exists (during the period of application of the brakes) a
+gravitational field which is directed forwards and which is variable
+with respect to time. Under the influence of this field, the embankment
+together with the earth moves non-uniformly in such a manner that their
+original velocity in the backwards direction is continuously reduced.”
+
+
+XXI.
+
+IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
+SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
+
+We have already stated several times that classical mechanics starts
+out from the following law: Material particles sufficiently far removed
+from other material particles continue to move uniformly in a straight
+line or continue in a state of rest. We have also repeatedly emphasised
+that this fundamental law can only be valid for bodies of reference _K_
+which possess certain unique states of motion, and which are in uniform
+translational motion relative to each other. Relative to other
+reference-bodies _K_ the law is not valid. Both in classical mechanics
+and in the special theory of relativity we therefore differentiate
+between reference-bodies _K_ relative to which the recognised “laws of
+nature” can be said to hold, and reference-bodies _K_ relative to which
+these laws do not hold.
+
+But no person whose mode of thought is logical can rest satisfied with
+this condition of things. He asks: “How does it come that certain
+reference-bodies (or their states of motion) are given priority over
+other reference-bodies (or their states of motion)? _What is the reason
+for this preference?_” In order to show clearly what I mean by this
+question, I shall make use of a comparison.
+
+I am standing in front of a gas range. Standing alongside of each other
+on the range are two pans so much alike that one may be mistaken for
+the other. Both are half full of water. I notice that steam is being
+emitted continuously from the one pan, but not from the other. I am
+surprised at this, even if I have never seen either a gas range or a
+pan before. But if I now notice a luminous something of bluish colour
+under the first pan but not under the other, I cease to be astonished,
+even if I have never before seen a gas flame. For I can only say that
+this bluish something will cause the emission of the steam, or at least
+_possibly_ it may do so. If, however, I notice the bluish something in
+neither case, and if I observe that the one continuously emits steam
+whilst the other does not, then I shall remain astonished and
+dissatisfied until I have discovered some circumstance to which I can
+attribute the different behaviour of the two pans.
+
+Analogously, I seek in vain for a real something in classical mechanics
+(or in the special theory of relativity) to which I can attribute the
+different behaviour of bodies considered with respect to the reference
+systems _K_ and _K′_.[15] Newton saw this objection and attempted to
+invalidate it, but without success. But E. Mach recognised it most
+clearly of all, and because of this objection he claimed that mechanics
+must be placed on a new basis. It can only be got rid of by means of a
+physics which is conformable to the general principle of relativity,
+since the equations of such a theory hold for every body of reference,
+whatever may be its state of motion.
+
+
+ [15] The objection is of importance more especially when the state of
+ motion of the reference-body is of such a nature that it does not
+ require any external agency for its maintenance, _e.g._ in the case
+ when the reference-body is rotating uniformly.
+
+
+XXII.
+
+A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
+
+The considerations of Section XX show that the general principle of
+relativity puts us in a position to derive properties of the
+gravitational field in a purely theoretical manner. Let us suppose, for
+instance, that we know the space-time “course” for any natural process
+whatsoever, as regards the manner in which it takes place in the
+Galileian domain relative to a Galileian body of reference _K_. By
+means of purely theoretical operations (_i.e._ simply by calculation)
+we are then able to find how this known natural process appears, as
+seen from a reference-body _K′_ which is accelerated relatively to _K_.
+But since a gravitational field exists with respect to this new body of
+reference _K′_, our consideration also teaches us how the gravitational
+field influences the process studied.
+
+For example, we learn that a body which is in a state of uniform
+rectilinear motion with respect to _K_ (in accordance with the law of
+Galilei) is executing an accelerated and in general curvilinear motion
+with respect to the accelerated reference-body _K′_ (chest). This
+acceleration or curvature corresponds to the influence on the moving
+body of the gravitational field prevailing relatively to _K_. It is
+known that a gravitational field influences the movement of bodies in
+this way, so that our consideration supplies us with nothing
+essentially new.
+
+However, we obtain a new result of fundamental importance when we carry
+out the analogous consideration for a ray of light. With respect to the
+Galileian reference-body _K_, such a ray of light is transmitted
+rectilinearly with the velocity _c_. It can easily be shown that the
+path of the same ray of light is no longer a straight line when we
+consider it with reference to the accelerated chest (reference-body
+_K′_). From this we conclude, _that, in general, rays of light are
+propagated curvilinearly in gravitational fields._ In two respects this
+result is of great importance.
+
+In the first place, it can be compared with the reality. Although a
+detailed examination of the question shows that the curvature of light
+rays required by the general theory of relativity is only exceedingly
+small for the gravitational fields at our disposal in practice, its
+estimated magnitude for light rays passing the sun at grazing incidence
+is nevertheless 1.7 seconds of arc. This ought to manifest itself in
+the following way. As seen from the earth, certain fixed stars appear
+to be in the neighbourhood of the sun, and are thus capable of
+observation during a total eclipse of the sun. At such times, these
+stars ought to appear to be displaced outwards from the sun by an
+amount indicated above, as compared with their apparent position in the
+sky when the sun is situated at another part of the heavens. The
+examination of the correctness or otherwise of this deduction is a
+problem of the greatest importance, the early solution of which is to
+be expected of astronomers.[16]
+
+
+ [16] By means of the star photographs of two expeditions equipped by a
+ Joint Committee of the Royal and Royal Astronomical Societies, the
+ existence of the deflection of light demanded by theory was first
+ confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
+ III.)
+
+
+In the second place our result shows that, according to the general
+theory of relativity, the law of the constancy of the velocity of light
+in vacuo, which constitutes one of the two fundamental assumptions in
+the special theory of relativity and to which we have already
+frequently referred, cannot claim any unlimited validity. A curvature
+of rays of light can only take place when the velocity of propagation
+of light varies with position. Now we might think that as a consequence
+of this, the special theory of relativity and with it the whole theory
+of relativity would be laid in the dust. But in reality this is not the
+case. We can only conclude that the special theory of relativity cannot
+claim an unlimited domain of validity; its results hold only so long as
+we are able to disregard the influences of gravitational fields on the
+phenomena (_e.g._ of light).
+
+Since it has often been contended by opponents of the theory of
+relativity that the special theory of relativity is overthrown by the
+general theory of relativity, it is perhaps advisable to make the facts
+of the case clearer by means of an appropriate comparison. Before the
+development of electrodynamics the laws of electrostatics were looked
+upon as the laws of electricity. At the present time we know that
+electric fields can be derived correctly from electrostatic
+considerations only for the case, which is never strictly realised, in
+which the electrical masses are quite at rest relatively to each other,
+and to the co-ordinate system. Should we be justified in saying that
+for this reason electrostatics is overthrown by the field-equations of
+Maxwell in electrodynamics? Not in the least. Electrostatics is
+contained in electrodynamics as a limiting case; the laws of the latter
+lead directly to those of the former for the case in which the fields
+are invariable with regard to time. No fairer destiny could be allotted
+to any physical theory, than that it should of itself point out the way
+to the introduction of a more comprehensive theory, in which it lives
+on as a limiting case.
+
+In the example of the transmission of light just dealt with, we have
+seen that the general theory of relativity enables us to derive
+theoretically the influence of a gravitational field on the course of
+natural processes, the laws of which are already known when a
+gravitational field is absent. But the most attractive problem, to the
+solution of which the general theory of relativity supplies the key,
+concerns the investigation of the laws satisfied by the gravitational
+field itself. Let us consider this for a moment.
+
+We are acquainted with space-time domains which behave (approximately)
+in a “Galileian” fashion under suitable choice of reference-body,
+_i.e._ domains in which gravitational fields are absent. If we now
+refer such a domain to a reference-body _K′_ possessing any kind of
+motion, then relative to _K′_ there exists a gravitational field which
+is variable with respect to space and time.[17] The character of this
+field will of course depend on the motion chosen for _K′._ According to
+the general theory of relativity, the general law of the gravitational
+field must be satisfied for all gravitational fields obtainable in this
+way. Even though by no means all gravitationial fields can be produced
+in this way, yet we may entertain the hope that the general law of
+gravitation will be derivable from such gravitational fields of a
+special kind. This hope has been realised in the most beautiful manner.
+But between the clear vision of this goal and its actual realisation it
+was necessary to surmount a serious difficulty, and as this lies deep
+at the root of things, I dare not withhold it from the reader. We
+require to extend our ideas of the space-time continuum still farther.
+
+
+ [17] This follows from a generalisation of the discussion in Section
+ XX.
+
+
+XXIII.
+
+BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
+
+Hitherto I have purposely refrained from speaking about the physical
+interpretation of space- and time-data in the case of the general
+theory of relativity. As a consequence, I am guilty of a certain
+slovenliness of treatment, which, as we know from the special theory of
+relativity, is far from being unimportant and pardonable. It is now
+high time that we remedy this defect; but I would mention at the
+outset, that this matter lays no small claims on the patience and on
+the power of abstraction of the reader.
+
+We start off again from quite special cases, which we have frequently
+used before. Let us consider a space time domain in which no
+gravitational field exists relative to a reference-body _K_ whose state
+of motion has been suitably chosen. _K_ is then a Galileian
+reference-body as regards the domain considered, and the results of the
+special theory of relativity hold relative to _K_. Let us suppose the
+same domain referred to a second body of reference _K′_, which is
+rotating uniformly with respect to _K_. In order to fix our ideas, we
+shall imagine _K′_ to be in the form of a plane circular disc, which
+rotates uniformly in its own plane about its centre. An observer who is
+sitting eccentrically on the disc _K′_ is sensible of a force which
+acts outwards in a radial direction, and which would be interpreted as
+an effect of inertia (centrifugal force) by an observer who was at rest
+with respect to the original reference-body _K_. But the observer on
+the disc may regard his disc as a reference-body which is “at rest”; on
+the basis of the general principle of relativity he is justified in
+doing this. The force acting on himself, and in fact on all other
+bodies which are at rest relative to the disc, he regards as the effect
+of a gravitational field. Nevertheless, the space-distribution of this
+gravitational field is of a kind that would not be possible on Newton’s
+theory of gravitation.[18] But since the observer believes in the
+general theory of relativity, this does not disturb him; he is quite in
+the right when he believes that a general law of gravitation can be
+formulated—a law which not only explains the motion of the stars
+correctly, but also the field of force experienced by himself.
+
+
+ [18] The field disappears at the centre of the disc and increases
+ proportionally to the distance from the centre as we proceed outwards.
+
+
+The observer performs experiments on his circular disc with clocks and
+measuring-rods. In doing so, it is his intention to arrive at exact
+definitions for the signification of time- and space-data with
+reference to the circular disc _K′_, these definitions being based on
+his observations. What will be his experience in this enterprise?
+
+To start with, he places one of two identically constructed clocks at
+the centre of the circular disc, and the other on the edge of the disc,
+so that they are at rest relative to it. We now ask ourselves whether
+both clocks go at the same rate from the standpoint of the non-rotating
+Galileian reference-body _K_. As judged from this body, the clock at
+the centre of the disc has no velocity, whereas the clock at the edge
+of the disc is in motion relative to _K_ in consequence of the
+rotation. According to a result obtained in Section XII, it follows
+that the latter clock goes at a rate permanently slower than that of
+the clock at the centre of the circular disc, _i.e._ as observed from
+_K_. It is obvious that the same effect would be noted by an observer
+whom we will imagine sitting alongside his clock at the centre of the
+circular disc. Thus on our circular disc, or, to make the case more
+general, in every gravitational field, a clock will go more quickly or
+less quickly, according to the position in which the clock is situated
+(at rest). For this reason it is not possible to obtain a reasonable
+definition of time with the aid of clocks which are arranged at rest
+with respect to the body of reference. A similar difficulty presents
+itself when we attempt to apply our earlier definition of simultaneity
+in such a case, but I do not wish to go any farther into this question.
+
+Moreover, at this stage the definition of the space co-ordinates also
+presents insurmountable difficulties. If the observer applies his
+standard measuring-rod (a rod which is short as compared with the
+radius of the disc) tangentially to the edge of the disc, then, as
+judged from the Galileian system, the length of this rod will be less
+than 1, since, according to Section XII, moving bodies suffer a
+shortening in the direction of the motion. On the other hand, the
+measuring-rod will not experience a shortening in length, as judged
+from _K_, if it is applied to the disc in the direction of the radius.
+If, then, the observer first measures the circumference of the disc
+with his measuring-rod and then the diameter of the disc, on dividing
+the one by the other, he will not obtain as quotient the familiar
+number π = 3.14 . . ., but a larger number,[19] whereas of course, for
+a disc which is at rest with respect to _K_, this operation would yield
+π exactly. This proves that the propositions of Euclidean geometry
+cannot hold exactly on the rotating disc, nor in general in a
+gravitational field, at least if we attribute the length 1 to the rod
+in all positions and in every orientation. Hence the idea of a straight
+line also loses its meaning. We are therefore not in a position to
+define exactly the co-ordinates _x, y, z_ relative to the disc by means
+of the method used in discussing the special theory, and as long as the
+co-ordinates and times of events have not been defined, we cannot
+assign an exact meaning to the natural laws in which these occur.
+
+
+ [19] Throughout this consideration we have to use the Galileian
+ (non-rotating) system _K_ as reference-body, since we may only assume
+ the validity of the results of the special theory of relativity
+ relative to _K_ (relative to _K′_ a gravitational field prevails).
+
+
+Thus all our previous conclusions based on general relativity would
+appear to be called in question. In reality we must make a subtle
+detour in order to be able to apply the postulate of general relativity
+exactly. I shall prepare the reader for this in the following
+paragraphs.
+
+
+XXIV.
+
+EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
+
+The surface of a marble table is spread out in front of me. I can get
+from any one point on this table to any other point by passing
+continuously from one point to a “neighbouring” one, and repeating this
+process a (large) number of times, or, in other words, by going from
+point to point without executing “jumps.” I am sure the reader will
+appreciate with sufficient clearness what I mean here by “neighbouring”
+and by “jumps” (if he is not too pedantic). We express this property of
+the surface by describing the latter as a continuum.
+
+Let us now imagine that a large number of little rods of equal length
+have been made, their lengths being small compared with the dimensions
+of the marble slab. When I say they are of equal length, I mean that
+one can be laid on any other without the ends overlapping. We next lay
+four of these little rods on the marble slab so that they constitute a
+quadrilateral figure (a square), the diagonals of which are equally
+long. To ensure the equality of the diagonals, we make use of a little
+testing-rod. To this square we add similar ones, each of which has one
+rod in common with the first. We proceed in like manner with each of
+these squares until finally the whole marble slab is laid out with
+squares. The arrangement is such, that each side of a square belongs to
+two squares and each corner to four squares.
+
+It is a veritable wonder that we can carry out this business without
+getting into the greatest difficulties. We only need to think of the
+following. If at any moment three squares meet at a corner, then two
+sides of the fourth square are already laid, and, as a consequence, the
+arrangement of the remaining two sides of the square is already
+completely determined. But I am now no longer able to adjust the
+quadrilateral so that its diagonals may be equal. If they are equal of
+their own accord, then this is an especial favour of the marble slab
+and of the little rods, about which I can only be thankfully surprised.
+We must experience many such surprises if the construction is to be
+successful.
+
+If everything has really gone smoothly, then I say that the points of
+the marble slab constitute a Euclidean continuum with respect to the
+little rod, which has been used as a “distance” (line-interval). By
+choosing one corner of a square as “origin” I can characterise every
+other corner of a square with reference to this origin by means of two
+numbers. I only need state how many rods I must pass over when,
+starting from the origin, I proceed towards the “right” and then
+“upwards,” in order to arrive at the corner of the square under
+consideration. These two numbers are then the “Cartesian co-ordinates”
+of this corner with reference to the “Cartesian co-ordinate system”
+which is determined by the arrangement of little rods.
+
+By making use of the following modification of this abstract
+experiment, we recognise that there must also be cases in which the
+experiment would be unsuccessful. We shall suppose that the rods
+“expand” by in amount proportional to the increase of temperature. We
+heat the central part of the marble slab, but not the periphery, in
+which case two of our little rods can still be brought into coincidence
+at every position on the table. But our construction of squares must
+necessarily come into disorder during the heating, because the little
+rods on the central region of the table expand, whereas those on the
+outer part do not.
+
+With reference to our little rods—defined as unit lengths—the marble
+slab is no longer a Euclidean continuum, and we are also no longer in
+the position of defining Cartesian co-ordinates directly with their
+aid, since the above construction can no longer be carried out. But
+since there are other things which are not influenced in a similar
+manner to the little rods (or perhaps not at all) by the temperature of
+the table, it is possible quite naturally to maintain the point of view
+that the marble slab is a “Euclidean continuum.” This can be done in a
+satisfactory manner by making a more subtle stipulation about the
+measurement or the comparison of lengths.
+
+But if rods of every kind (_i.e._ of every material) were to behave _in
+the same way_ as regards the influence of temperature when they are on
+the variably heated marble slab, and if we had no other means of
+detecting the effect of temperature than the geometrical behaviour of
+our rods in experiments analogous to the one described above, then our
+best plan would be to assign the distance one to two points on the
+slab, provided that the ends of one of our rods could be made to
+coincide with these two points; for how else should we define the
+distance without our proceeding being in the highest measure grossly
+arbitrary? The method of Cartesian coordinates must then be discarded,
+and replaced by another which does not assume the validity of Euclidean
+geometry for rigid bodies.[20] The reader will notice that the
+situation depicted here corresponds to the one brought about by the
+general postulate of relativity (Section XXIII).
+
+
+ [20] Mathematicians have been confronted with our problem in the
+ following form. If we are given a surface (_e.g._ an ellipsoid) in
+ Euclidean three-dimensional space, then there exists for this surface
+ a two-dimensional geometry, just as much as for a plane surface. Gauss
+ undertook the task of treating this two-dimensional geometry from
+ first principles, without making use of the fact that the surface
+ belongs to a Euclidean continuum of three dimensions. If we imagine
+ constructions to be made with rigid rods _in the surface_ (similar to
+ that above with the marble slab), we should find that different laws
+ hold for these from those resulting on the basis of Euclidean plane
+ geometry. The surface is not a Euclidean continuum with respect to the
+ rods, and we cannot define Cartesian co-ordinates _in the surface_.
+ Gauss indicated the principles according to which we can treat the
+ geometrical relationships in the surface, and thus pointed out the way
+ to the method of Riemann of treating multi-dimensional, non-Euclidean
+ _continuum_. Thus it is that mathematicians long ago solved the formal
+ problems to which we are led by the general postulate of relativity.
+
+
+XXV.
+
+GAUSSIAN CO-ORDINATES
+
+image033
+
+
+According to Gauss, this combined analytical and geometrical mode of
+handling the problem can be arrived at in the following way. We imagine
+a system of arbitrary curves (see Fig. 4) drawn on the surface of the
+table. These we designate as _u_-curves, and we indicate each of them
+by means of a number. The Curves _u_ = 1, _u_ = 2 and _u_ = 3 are drawn
+in the diagram. Between the curves _u_ = 1 and _u_ = 2 we must imagine
+an infinitely large number to be drawn, all of which correspond to real
+numbers lying between 1 and 2. We have then a system of _u_-curves, and
+this “infinitely dense” system covers the whole surface of the table.
+These _u_-curves must not intersect each other, and through each point
+of the surface one and only one curve must pass. Thus a perfectly
+definite value of _u_ belongs to every point on the surface of the
+marble slab. In like manner we imagine a system of _v_-curves drawn on
+the surface. These satisfy the same conditions as the _u_-curves, they
+are provided with numbers in a corresponding manner, and they may
+likewise be of arbitrary shape. It follows that a value of _u_ and a
+value of _v_ belong to every point on the surface of the table. We call
+these two numbers the co-ordinates of the surface of the table
+(Gaussian co-ordinates). For example, the point _P_ in the diagram has
+the Gaussian co-ordinates _u_ = 3, _v_ = 1. Two neighbouring points _P_
+and _P′_ on the surface then correspond to the co-ordinates
+
+_P_: _u, v_
+
+_P′_: _u_ + _du, v_ + _dv_,
+
+where _du_ and _dv_ signify very small numbers. In a similar manner we
+may indicate the distance (line-interval) between _P_ and _P′_, as
+measured with a little rod, by means of the very small number _ds_.
+Then according to Gauss we have
+
+_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2,
+
+where _g_11, _g_12, _g_22, are magnitudes which depend in a perfectly
+definite way on _u_ and _v_. The magnitudes _g_11, _g_12 and _g_22,
+determine the behaviour of the rods relative to the _u_-curves and
+_v_-curves, and thus also relative to the surface of the table. For the
+case in which the points of the surface considered form a Euclidean
+continuum with reference to the measuring-rods, but only in this case,
+it is possible to draw the _u_-curves and _v_-curves and to attach
+numbers to them, in such a manner, that we simply have:
+
+_ds_2 = _du_2 + _dv_2
+
+Under these conditions, the _u_-curves and _v_-curves are straight
+lines in the sense of Euclidean geometry, and they are perpendicular to
+each other. Here the Gaussian coordinates are simply Cartesian ones. It
+is clear that Gauss co-ordinates are nothing more than an association
+of two sets of numbers with the points of the surface considered, of
+such a nature that numerical values differing very slightly from each
+other are associated with neighbouring points “in space.”
+
+So far, these considerations hold for a continuum of two dimensions.
+But the Gaussian method can be applied also to a continuum of three,
+four or more dimensions. If, for instance, a continuum of four
+dimensions be supposed available, we may represent it in the following
+way. With every point of the continuum, we associate arbitrarily four
+numbers, _x_1, _x_2, _x_3, _x_4, which are known as “co-ordinates.”
+Adjacent points correspond to adjacent values of the coordinates. If a
+distance _ds_ is associated with the adjacent points _P_ and _P′_, this
+distance being measurable and well defined from a physical point of
+view, then the following formula holds:
+
+_ds_2 = _g_11_dx_12 + 2_g_12_dx_1_dx_2 . . . . + _g_44_dx_42,
+
+where the magnitudes _g_11, etc., have values which vary with the
+position in the continuum. Only when the continuum is a Euclidean one
+is it possible to associate the co-ordinates _x_1 . . _x_4. with the
+points of the continuum so that we have simply
+
+_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
+
+In this case relations hold in the four-dimensional continuum which are
+analogous to those holding in our three-dimensional measurements.
+
+However, the Gauss treatment for _ds_2 which we have given above is not
+always possible. It is only possible when sufficiently small regions of
+the continuum under consideration may be regarded as Euclidean
+continua. For example, this obviously holds in the case of the marble
+slab of the table and local variation of temperature. The temperature
+is practically constant for a small part of the slab, and thus the
+geometrical behaviour of the rods is _almost_ as it ought to be
+according to the rules of Euclidean geometry. Hence the imperfections
+of the construction of squares in the previous section do not show
+themselves clearly until this construction is extended over a
+considerable portion of the surface of the table.
+
+We can sum this up as follows: Gauss invented a method for the
+mathematical treatment of continua in general, in which
+“size-relations” (“distances” between neighbouring points) are defined.
+To every point of a continuum are assigned as many numbers (Gaussian
+coordinates) as the continuum has dimensions. This is done in such a
+way, that only one meaning can be attached to the assignment, and that
+numbers (Gaussian coordinates) which differ by an indefinitely small
+amount are assigned to adjacent points. The Gaussian coordinate system
+is a logical generalisation of the Cartesian co-ordinate system. It is
+also applicable to non-Euclidean continua, but only when, with respect
+to the defined “size” or “distance,” small parts of the continuum under
+consideration behave more nearly like a Euclidean system, the smaller
+the part of the continuum under our notice.
+
+
+XXVI.
+
+THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED
+AS A EUCLIDEAN CONTINUUM
+
+We are now in a position to formulate more exactly the idea of
+Minkowski, which was only vaguely indicated in Section XVII. In
+accordance with the special theory of relativity, certain co-ordinate
+systems are given preference for the description of the
+four-dimensional, space-time continuum. We called these “Galileian
+co-ordinate systems.” For these systems, the four co-ordinates _x, y,
+z, t_, which determine an event or—in other words—a point of the
+four-dimensional continuum, are defined physically in a simple manner,
+as set forth in detail in the first part of this book. For the
+transition from one Galileian system to another, which is moving
+uniformly with reference to the first, the equations of the Lorentz
+transformation are valid. These last form the basis for the derivation
+of deductions from the special theory of relativity, and in themselves
+they are nothing more than the expression of the universal validity of
+the law of transmission of light for all Galileian systems of
+reference.
+
+Minkowski found that the Lorentz transformations satisfy the following
+simple conditions. Let us consider two neighbouring events, the
+relative position of which in the four-dimensional continuum is given
+with respect to a Galileian reference-body _K_ by the space co-ordinate
+differences _dx, dy, dz_ and the time-difference _dt_. With reference
+to a second Galileian system we shall suppose that the corresponding
+differences for these two events are _dx′, dy′, dz′, dt′_. Then these
+magnitudes always fulfill the condition.[21]
+
+
+ [21] Cf. Appendixes I and II. The relations which are derived there
+ for the co-ordinates themselves are valid also for co-ordinate
+ _differences_, and thus also for co-ordinate differentials
+ (indefinitely small differences).
+
+
+_dx_2 + _dy_2 + _dz_2 – _c_2_dt_2 = _dx′_2 + _dy′_2 + _dz′_2 –
+_c_2_dt′_2.
+
+
+The validity of the Lorentz transformation follows from this condition.
+We can express this as follows: The magnitude
+
+_ds_2 = _dx_2 + _dy_2 + _dz_2 – _c_2 _dt_2,
+
+
+which belongs to two adjacent points of the four-dimensional space-time
+continuum, has the same value for all selected (Galileian)
+reference-bodies. If we replace _x, y, z_,
+
+image034
+
+
+by _x_1, _x_2, _x_3, _x_4, we also obtain the result that
+
+_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
+
+
+is independent of the choice of the body of reference. We call the
+magnitude _ds_ the “distance” apart of the two events or
+four-dimensional points.
+
+Thus, if we choose as time-variable the imaginary variable
+
+image035
+
+
+instead of the real quantity _t_, we can regard the space-time
+contintium—accordance with the special theory of relativity—as a
+“Euclidean” four-dimensional continuum, a result which follows from the
+considerations of the preceding section.
+
+
+XXVII.
+
+THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A
+EUCLIDEAN CONTINUUM
+
+In the first part of this book we were able to make use of space-time
+co-ordinates which allowed of a simple and direct physical
+interpretation, and which, according to Section XXVI, can be regarded
+as four-dimensional Cartesian co-ordinates. This was possible on the
+basis of the law of the constancy of the velocity of light. But
+according to Section XXI the general theory of relativity cannot retain
+this law. On the contrary, we arrived at the result that according to
+this latter theory the velocity of light must always depend on the
+co-ordinates when a gravitational field is present. In connection with
+a specific illustration in Section XXIII, we found that the presence of
+a gravitational field invalidates the definition of the coordinates and
+the time, which led us to our objective in the special theory of
+relativity.
+
+In view of the resuIts of these considerations we are led to the
+conviction that, according to the general principle of relativity, the
+space-time continuum cannot be regarded as a Euclidean one, but that
+here we have the general case, corresponding to the marble slab with
+local variations of temperature, and with which we made acquaintance as
+an example of a two-dimensional continuum. Just as it was there
+impossible to construct a Cartesian co-ordinate system from equal rods,
+so here it is impossible to build up a system (reference-body) from
+rigid bodies and clocks, which shall be of such a nature that
+measuring-rods and clocks, arranged rigidly with respect to one
+another, shall indicate position and time directly. Such was the
+essence of the difficulty with which we were confronted in Section
+XXIII.
+
+But the considerations of Sections XXV and XXVI show us the way to
+surmount this difficulty. We refer the four-dimensional space-time
+continuum in an arbitrary manner to Gauss co-ordinates. We assign to
+every point of the continuum (event) four numbers, _x_1, _x_2, _x_3,
+_x_4 (co-ordinates), which have not the least direct physical
+significance, but only serve the purpose of numbering the points of the
+continuum in a definite but arbitrary manner. This arrangement does not
+even need to be of such a kind that we must regard _x_1, _x_2, _x_3, as
+“space” co-ordinates and _x_4, as a “time” co-ordinate.
+
+The reader may think that such a description of the world would be
+quite inadequate. What does it mean to assign to an event the
+particular co-ordinates _x_1, _x_2, _x_3, _x_4, if in themselves these
+co-ordinates have no significance? More careful consideration shows,
+however, that this anxiety is unfounded. Let us consider, for instance,
+a material point with any kind of motion. If this point had only a
+momentary existence without duration, then it would to described in
+space-time by a single system of values _x_1, _x_2, _x_3, _x_4. Thus
+its permanent existence must be characterised by an infinitely large
+number of such systems of values, the co-ordinate values of which are
+so close together as to give continuity; corresponding to the material
+point, we thus have a (uni-dimensional) line in the four-dimensional
+continuum. In the same way, any such lines in our continuum correspond
+to many points in motion. The only statements having regard to these
+points which can claim a physical existence are in reality the
+statements about their encounters. In our mathematical treatment, such
+an encounter is expressed in the fact that the two lines which
+represent the motions of the points in question have a particular
+system of co-ordinate values, _x_1, _x_2, _x_3, _x_4, in common. After
+mature consideration the reader will doubtless admit that in reality
+such encounters constitute the only actual evidence of a time-space
+nature with which we meet in physical statements.
+
+When we were describing the motion of a material point relative to a
+body of reference, we stated nothing more than the encounters of this
+point with particular points of the reference-body. We can also
+determine the corresponding values of the time by the observation of
+encounters of the body with clocks, in conjunction with the observation
+of the encounter of the hands of clocks with particular points on the
+dials. It is just the same in the case of space-measurements by means
+of measuring-rods, as a little consideration will show.
+
+The following statements hold generally: Every physical description
+resolves itself into a number of statements, each of which refers to
+the space-time coincidence of two events _A_ and _B_. In terms of
+Gaussian co-ordinates, every such statement is expressed by the
+agreement of their four co-ordinates _x_1, _x_2, _x_3, _x_4. Thus in
+reality, the description of the time-space continuum by means of Gauss
+co-ordinates completely replaces the description with the aid of a body
+of reference, without suffering from the defects of the latter mode of
+description; it is not tied down to the Euclidean character of the
+continuum which has to be represented.
+
+
+XXVIII.
+
+EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
+
+We are now in a position to replace the provisional formulation of the
+general principle of relativity given in Section XVIII by an exact
+formulation. The form there used, “All bodies of reference _K, K′_,
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion,” cannot be maintained, because the use of rigid
+reference-bodies, in the sense of the method followed in the special
+theory of relativity, is in general not possible in space-time
+description. The Gauss co-ordinate system has to take the place of the
+body of reference. The following statement corresponds to the
+fundamental idea of the general principle of relativity: “_All Gaussian
+co-ordinate systems are essentially equivalent for the formulation of
+the general laws of nature._”
+
+We can state this general principle of relativity in still another
+form, which renders it yet more clearly intelligible than it is when in
+the form of the natural extension of the special principle of
+relativity. According to the special theory of relativity, the
+equations which express the general laws of nature pass over into
+equations of the same form when, by making use of the Lorentz
+transformation, we replace the space-time variables _x, y, z, t_, of a
+(Galileian) reference-body _K_ by the space-time variables _x′, y′, z′,
+t′_, of a new reference-body _K′_. According to the general theory of
+relativity, on the other hand, by application of _arbitrary
+substitutions_ of the Gauss variables _x_1, _x_2, _x_3, _x_4, the
+equations must pass over into equations of the same form; for every
+transformation (not only the Lorentz transformation) corresponds to the
+transition of one Gauss co-ordinate system into another.
+
+If we desire to adhere to our “old-time” three-dimensional view of
+things, then we can characterise the development which is being
+undergone by the fundamental idea of the general theory of relativity
+as follows: The special theory of relativity has reference to Galileian
+domains, _i.e._ to those in which no gravitational field exists. In
+this connection a Galileian reference-body serves as body of reference,
+_i.e._ a rigid body the state of motion of which is so chosen that the
+Galileian law of the uniform rectilinear motion of “isolated” material
+points holds relatively to it.
+
+Certain considerations suggest that we should refer the same Galileian
+domains to _non-Galileian_ reference-bodies also. A gravitational field
+of a special kind is then present with respect to these bodies (cf.
+Sections XX and XXIII).
+
+In gravitational fields there are no such things as rigid bodies with
+Euclidean properties; thus the fictitious rigid body of reference is of
+no avail in the general theory of relativity. The motion of clocks is
+also influenced by gravitational fields, and in such a way that a
+physical definition of time which is made directly with the aid of
+clocks has by no means the same degree of plausibility as in the
+special theory of relativity.
+
+For this reason non-rigid reference-bodies are used, which are as a
+whole not only moving in any way whatsoever, but which also suffer
+alterations in form _ad lib._ during their motion. Clocks, for which
+the law of motion is of any kind, however irregular, serve for the
+definition of time. We have to imagine each of these clocks fixed at a
+point on the non-rigid reference-body. These clocks satisfy only the
+one condition, that the “readings” which are observed simultaneously on
+adjacent clocks (in space) differ from each other by an indefinitely
+small amount. This non-rigid reference-body, which might appropriately
+be termed a “reference-mollusc”, is in the main equivalent to a
+Gaussian four-dimensional co-ordinate system chosen arbitrarily. That
+which gives the “mollusc” a certain comprehensibility as compared with
+the Gauss co-ordinate system is the (really unjustified) formal
+retention of the separate existence of the
+
+space co-ordinates as opposed to the time co-ordinate. Every point on
+the mollusc is treated as a space-point, and every material point which
+is at rest relatively to it as at rest, so long as the mollusc is
+considered as reference-body. The general principle of relativity
+requires that all these molluscs can be used as reference-bodies with
+equal right and equal success in the formulation of the general laws of
+nature; the laws themselves must be quite independent of the choice of
+mollusc.
+
+The great power possessed by the general principle of relativity lies
+in the comprehensive limitation which is imposed on the laws of nature
+in consequence of what we have seen above.
+
+
+XXIX.
+
+THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
+PRINCIPLE OF RELATIVITY
+
+If the reader has followed all our previous considerations, he will
+have no further difficulty in understanding the methods leading to the
+solution of the problem of gravitation.
+
+We start off on a consideration of a Galileian domain, _i.e._ a domain
+in which there is no gravitational field relative to the Galileian
+reference-body _K_. The behaviour of measuring-rods and clocks with
+reference to _K_ is known from the special theory of relativity,
+likewise the behaviour of “isolated” material points; the latter move
+uniformly and in straight lines.
+
+Now let us refer this domain to a random Gauss coordinate system or to
+a “mollusc” as reference-body _K′_. Then with respect to _K′_ there is
+a gravitational field _G_ (of a particular kind). We learn the
+behaviour of measuring-rods and clocks and also of freely-moving
+material points with reference to _K′_ simply by mathematical
+transformation. We interpret this behaviour as the behaviour of
+measuring-rods, clocks and material points under the influence of the
+gravitational field _G_. Hereupon we introduce a hypothesis: that the
+influence of the gravitational field on measuring-rods, clocks and
+freely-moving material points continues to take place according to the
+same laws, even in the case where the prevailing gravitational field is
+_not_ derivable from the Galileian special case, simply by means of a
+transformation of co-ordinates.
+
+The next step is to investigate the space-time behaviour of the
+gravitational field _G_, which was derived from the Galileian special
+case simply by transformation of the coordinates. This behaviour is
+formulated in a law, which is always valid, no matter how the
+reference-body (mollusc) used in the description may be chosen.
+
+This law is not yet the _general_ law of the gravitational field, since
+the gravitational field under consideration is of a special kind. In
+order to find out the general law-of-field of gravitation we still
+require to obtain a generalisation of the law as found above. This can
+be obtained without caprice, however, by taking into consideration the
+following demands:
+
+(_a_) The required generalisation must likewise satisfy the general
+postulate of relativity.
+
+
+(_b_) If there is any matter in the domain under consideration, only
+its inertial mass, and thus according to Section XV only its energy is
+of importance for its effect in exciting a field.
+
+
+(_c_) Gravitational field and matter together must satisfy the law of
+the conservation of energy (and of impulse).
+
+
+Finally, the general principle of relativity permits us to determine
+the influence of the gravitational field on the course of all those
+processes which take place according to known laws when a gravitational
+field is absent _i.e._ which have already been fitted into the frame of
+the special theory of relativity. In this connection we proceed in
+principle according to the method which has already been explained for
+measuring-rods, clocks and freely moving material points.
+
+The theory of gravitation derived in this way from the general
+postulate of relativity excels not only in its beauty; nor in removing
+the defect attaching to classical mechanics which was brought to light
+in Section XXI; nor in interpreting the empirical law of the equality
+of inertial and gravitational mass; but it has also already explained a
+result of observation in astronomy, against which classical mechanics
+is powerless.
+
+If we confine the application of the theory to the case where the
+gravitational fields can be regarded as being weak, and in which all
+masses move with respect to the coordinate system with velocities which
+are small compared with the velocity of light, we then obtain as a
+first approximation the Newtonian theory. Thus the latter theory is
+obtained here without any particular assumption, whereas Newton had to
+introduce the hypothesis that the force of attraction between mutually
+attracting material points is inversely proportional to the square of
+the distance between them. If we increase the accuracy of the
+calculation, deviations from the theory of Newton make their
+appearance, practically all of which must nevertheless escape the test
+of observation owing to their smallness.
+
+We must draw attention here to one of these deviations. According to
+Newton’s theory, a planet moves round the sun in an ellipse, which
+would permanently maintain its position with respect to the fixed
+stars, if we could disregard the motion of the fixed stars themselves
+and the action of the other planets under consideration. Thus, if we
+correct the observed motion of the planets for these two influences,
+and if Newton’s theory be strictly correct, we ought to obtain for the
+orbit of the planet an ellipse, which is fixed with reference to the
+fixed stars. This deduction, which can be tested with great accuracy,
+has been confirmed for all the planets save one, with the precision
+that is capable of being obtained by the delicacy of observation
+attainable at the present time. The sole exception is Mercury, the
+planet which lies nearest the sun. Since the time of Leverrier, it has
+been known that the ellipse corresponding to the orbit of Mercury,
+after it has been corrected for the influences mentioned above, is not
+stationary with respect to the fixed stars, but that it rotates
+exceedingly slowly in the plane of the orbit and in the sense of the
+orbital motion. The value obtained for this rotary movement of the
+orbital ellipse was 43 seconds of arc per century, an amount ensured to
+be correct to within a few seconds of arc. This effect can be explained
+by means of classical mechanics only on the assumption of hypotheses
+which have little probability, and which were devised solely for this
+purponse.
+
+On the basis of the general theory of relativity, it is found that the
+ellipse of every planet round the sun must necessarily rotate in the
+manner indicated above; that for all the planets, with the exception of
+Mercury, this rotation is too small to be detected with the delicacy of
+observation possible at the present time; but that in the case of
+Mercury it must amount to 43 seconds of arc per century, a result which
+is strictly in agreement with observation.
+
+Apart from this one, it has hitherto been possible to make only two
+deductions from the theory which admit of being tested by observation,
+to wit, the curvature of light rays by the gravitational field of the
+sun,[22] and a displacement of the spectral lines of light reaching us
+from large stars, as compared with the corresponding lines for light
+produced in an analogous manner terrestrially (_i.e._ by the same kind
+of atom).[23] These two deductions from the theory have both been
+confirmed.
+
+
+ [22] First observed by Eddington and others in 1919. (Cf. Appendix
+ III).
+
+
+ [23] Established by Adams in 1924. (Cf. p. 132)
+
+
+PART III: CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
+
+
+XXX.
+
+COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY
+
+Part from the difficulty discussed in Section XXI, there is a second
+fundamental difficulty attending classical celestial mechanics, which,
+to the best of my knowledge, was first discussed in detail by the
+astronomer Seeliger. If we ponder over the question as to how the
+universe, considered as a whole, is to be regarded, the first answer
+that suggests itself to us is surely this: As regards space (and time)
+the universe is infinite. There are stars everywhere, so that the
+density of matter, although very variable in detail, is nevertheless on
+the average everywhere the same. In other words: However far we might
+travel through space, we should find everywhere an attenuated swarm of
+fixed stars of approrimately the same kind and density.
+
+This view is not in harmony with the theory of Newton. The latter
+theory rather requires that the universe should have a kind of centre
+in which the density of the stars is a maximum, and that as we proceed
+outwards from this centre the group-density of the stars should
+diminish, until finally, at great distances, it is succeeded by an
+infinite region of emptiness. The stellar universe ought to be a finite
+island in the infinite ocean of space.[24]
+
+
+ [24] _Proof_—According to the theory of Newton, the number of “lines
+ of force” which come from infinity and terminate in a mass m is
+ proportional to the mass _m_. If, on the average, the mass density ρ0
+ is constant throughout the universe, then a sphere of volume _V_ will
+ enclose the average mass ρ0_V_. Thus the number of lines of force
+ passing through the surface _F_ of the sphere into its interior is
+ proportional to ρ0_V_. For unit area of the surface of the sphere the
+ number of lines of force which enters the sphere is thus proportional
+ to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the
+ surface would ultimately become infinite with increasing radius _R_ of
+ the sphere, which is impossible.
+
+
+This conception is in itself not very satisfactory. It is still less
+satisfactory because it leads to the result that the light emitted by
+the stars and also individual stars of the stellar system are
+perpetually passing out into infinite space, never to return, and
+without ever again coming into interaction with other objects of
+nature. Such a finite material universe would be destined to become
+gradually but systematically impoverished.
+
+In order to escape this dilemma, Seeliger suggested a modification of
+Newton’s law, in which he assumes that for great distances the force of
+attraction between two masses diminishes more rapidly than would result
+from the inverse square law. In this way it is possible for the mean
+density of matter to be constant everywhere, even to infinity, without
+infinitely large gravitational fields being produced. We thus free
+ourselves from the distasteful conception that the material universe
+ought to possess something of the nature of a centre. Of course we
+purchase our emancipation from the fundamental difficulties mentioned,
+at the cost of a modification and complication of Newton’s law which
+has neither empirical nor theoretical foundation. We can imagine
+innumerable laws which would serve the same purpose, without our being
+able to state a reason why one of them is to be preferred to the
+others; for any one of these laws would be founded just as little on
+more general theoretical principles as is the law of Newton.
+
+
+XXXI.
+
+THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE
+
+But speculations on the structure of the universe also move in quite
+another direction. The development of non-Euclidean geometry led to the
+recognition of the fact, that we can cast doubt on the _infiniteness_
+of our space without coming into conflict with the laws of thought or
+with experience (Riemann, Helmholtz). These questions have already been
+treated in detail and with unsurpassable lucidity by Helmholtz and
+Poincaré, whereas I can only touch on them briefly here.
+
+In the first place, we imagine an existence in two dimensional space.
+Flat beings with flat implements, and in particular flat rigid
+measuring-rods, are free to move in a _plane_. For them nothing exists
+outside of this plane: that which they observe to happen to themselves
+and to their flat “things” is the all-inclusive reality of their plane.
+In particular, the constructions of plane Euclidean geometry can be
+carried out by means of the rods _e.g._ the lattice construction,
+considered in Section XXIV. In contrast to ours, the universe of these
+beings is two-dimensional; but, like ours, it extends to infinity. In
+their universe there is room for an infinite number of identical
+squares made up of rods, _i.e._ its volume (surface) is infinite. If
+these beings say their universe is “plane,” there is sense in the
+statement, because they mean that they can perform the constructions of
+plane Euclidean geometry with their rods. In this connection the
+individual rods always represent the same distance, independently of
+their position.
+
+Let us consider now a second two-dimensional existence, but this time
+on a spherical surface instead of on a plane. The flat beings with
+their measuring-rods and other objects fit exactly on this surface and
+they are unable to leave it. Their whole universe of observation
+extends exclusively over the surface of the sphere. Are these beings
+able to regard the geometry of their universe as being plane geometry
+and their rods withal as the realisation of “distance”? They cannot do
+this. For if they attempt to realise a straight line, they will obtain
+a curve, which we “three-dimensional beings” designate as a great
+circle, _i.e._ a self-contained line of definite finite length, which
+can be measured up by means of a measuring-rod. Similarly, this
+universe has a finite area that can be compared with the area, of a
+square constructed with rods. The great charm resulting from this
+consideration lies in the recognition of the fact that _the universe of
+these beings is finite and yet has no limits._
+
+But the spherical-surface beings do not need to go on a world-tour in
+order to perceive that they are not living in a Euclidean universe.
+They can convince themselves of this on every part of their “world,”
+provided they do not use too small a piece of it. Starting from a
+point, they draw “straight lines” (arcs of circles as judged in three
+dimensional space) of equal length in all directions. They will call
+the line joining the free ends of these lines a “circle.” For a plane
+surface, the ratio of the circumference of a circle to its diameter,
+both lengths being measured with the same rod, is, according to
+Euclidean geometry of the plane, equal to a constant value π, which is
+independent of the diameter of the circle. On their spherical surface
+our flat beings would find for this ratio the value
+
+image036
+
+
+_i.e._ a smaller value than π, the difference being the more
+considerable, the greater is the radius of the circle in comparison
+with the radius _R_ of the “world-sphere.” By means of this relation
+the spherical beings can determine the radius of their universe
+(“world”), even when only a relatively small part of their worldsphere
+is available for their measurements. But if this part is very small
+indeed, they will no longer be able to demonstrate that they are on a
+spherical “world” and not on a Euclidean plane, for a small part of a
+spherical surface differs only slightly from a piece of a plane of the
+same size.
+
+Thus if the spherical surface beings are living on a planet of which
+the solar system occupies only a negligibly small part of the spherical
+universe, they have no means of determining whether they are living in
+a finite or in an infinite universe, because the “piece of universe” to
+which they have access is in both cases practically plane, or
+Euclidean. It follows directly from this discussion, that for our
+sphere-beings the circumference of a circle first increases with the
+radius until the “circumference of the universe” is reached, and that
+it thenceforward gradually decreases to zero for still further
+increasing values of the radius. During this process the area of the
+circle continues to increase more and more, until finally it becomes
+equal to the total area of the whole “world-sphere.”
+
+Perhaps the reader will wonder why we have placed our “beings” on a
+sphere rather than on another closed surface. But this choice has its
+justification in the fact that, of all closed surfaces, the sphere is
+unique in possessing the property that all points on it are equivalent.
+I admit that the ratio of the circumference _c_ of a circle to its
+radius _r_ depends on _r_, but for a given value of _r_ it is the same
+for all points of the “worldsphere”; in other words, the “world-sphere”
+is a “surface of constant curvature.”
+
+To this two-dimensional sphere-universe there is a three-dimensional
+analogy, namely, the three-dimensional spherical space which was
+discovered by Riemann. its points are likewise all equivalent. It
+possesses a finite volume, which is determined by its “radius”
+(2π2_R_3). Is it possible to imagine a spherical space? To imagine a
+space means nothing else than that we imagine an epitome of our “space”
+experience, _i.e._ of experience that we can have in the movement of
+“rigid” bodies. In this sense we _can_ imagine a spherical space.
+
+Suppose we draw lines or stretch strings in all directions from a
+point, and mark off from each of these the distance _r_ with a
+measuring-rod. All the free end-points of these lengths lie on a
+spherical surface. We can specially measure up the area (_F_) of this
+surface by means of a square made up of measuring-rods. If the universe
+is Euclidean, then _F_ = 4π_r_2; if it is spherical, then _F_ is always
+less than 4π_r_2. With increasing values of _r, F_ increases from zero
+up to a maximum value which is determined by the “world-radius,” but
+for still further increasing values of _r_, the area gradually
+diminishes to zero. At first, the straight lines which radiate from the
+starting point diverge farther and farther from one another, but later
+they approach each other, and finally they run together again at a
+“counter-point” to the starting point. Under such conditions they have
+traversed the whole spherical space. It is easily seen that the
+three-dimensional spherical space is quite analogous to the
+two-dimensional spherical surface. It is finite (_i.e._ of finite
+volume), and has no bounds.
+
+It may be mentioned that there is yet another kind of curved space:
+“elliptical space.” It can be regarded as a curved space in which the
+two “counter-points” are identical (indistinguishable from each other).
+An elliptical universe can thus be considered to some extent as a
+curved universe possessing central symmetry.
+
+It follows from what has been said, that closed spaces without limits
+are conceivable. From amongst these, the spherical space (and the
+elliptical) excels in its simplicity, since all points on it are
+equivalent. As a result of this discussion, a most interesting question
+arises for astronomers and physicists, and that is whether the universe
+in which we live is infinite, or whether it is finite in the manner of
+the spherical universe. Our experience is far from being sufficient to
+enable us to answer this question. But the general theory of relativity
+permits of our answering it with a moderate degree of certainty, and in
+this connection the difficulty mentioned in Section XXX finds its
+solution.
+
+
+XXXII.
+
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
+
+According to the general theory of relativity, the geometrical
+properties of space are not independent, but they are determined by
+matter. Thus we can draw conclusions about the geometrical structure of
+the universe only if we base our considerations on the state of the
+matter as being something that is known. We know from experience that,
+for a suitably chosen co-ordinate system, the velocities of the stars
+are small as compared with the velocity of transmission of light. We
+can thus as a rough approximation arrive at a conclusion as to the
+nature of the universe as a whole, if we treat the matter as being at
+rest.
+
+We already know from our previous discussion that the behaviour of
+measuring-rods and clocks is influenced by gravitational fields, _i.e._
+by the distribution of matter. This in itself is sufficient to exclude
+the possibility of the exact validity of Euclidean geometry in our
+universe. But it is conceivable that our universe differs only slightly
+from a Euclidean one, and this notion seems all the more probable,
+since calculations show that the metrics of surrounding space is
+influenced only to an exceedingly small extent by masses even of the
+magnitude of our sun. We might imagine that, as regards geometry, our
+universe behaves analogously to a surface which is irregularly curved
+in its individual parts, but which nowhere departs appreciably from a
+plane: something like the rippled surface of a lake. Such a universe
+might fittingly be called a quasi-Euclidean universe. As regards its
+space it would be infinite. But calculation shows that in a
+quasi-Euclidean universe the average density of matter would
+necessarily be _nil_. Thus such a universe could not be inhabited by
+matter everywhere; it would present to us that unsatisfactory picture
+which we portrayed in Section XXX.
+
+If we are to have in the universe an average density of matter which
+differs from zero, however small may be that difference, then the
+universe cannot be quasi-Euclidean. On the contrary, the results of
+calculation indicate that if matter be distributed uniformly, the
+universe would necessarily be spherical (or elliptical). Since in
+reality the detailed distribution of matter is not uniform, the real
+universe will deviate in individual parts from the spherical, _i.e._
+the universe will be quasi-spherical. But it will be necessarily
+finite. In fact, the theory supplies us with a simple connection[25]
+between the space-expanse of the universe and the average density of
+matter in it.
+
+
+ [25] For the radius _R_ of the universe we obtain the equation
+
+
+image037
+
+
+The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027;
+ρ is the average density of the matter and _k_ is a constant connected
+with the Newtonian constant of gravitation.
+
+
+APPENDICES
+
+
+APPENDIX I
+
+SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
+
+(SUPPLEMENTARY TO SECTION XI)
+
+For the relative orientation of the co-ordinate systems indicated in
+Fig. 2, the _x_-axes of both systems permanently coincide. In the
+present case we can divide the problem into parts by considering first
+only events which are localised on the _x_-axis. Any such event is
+represented with respect to the co-ordinate system _K_ by the abscissa
+_x_ and the time _t_, and with respect to the system _K′_ by the
+abscissa _x′_ and the time _t′_. We require to find _x′_ and _t′_ when
+_x_ and _t_ are given.
+
+A light-signal, which is proceeding along the positive axis of _x_, is
+transmitted according to the equation
+
+_x_ = _ct_
+
+
+or
+
+_x_ – _ct_ = 0 . . . . . (1).
+
+
+Since the same light-signal has to be transmitted relative to _K′_ with
+the velocity _c_, the propagation relative to the system _K′_ will be
+represented by the analogous formula
+
+_x′_ – _ct′_ = 0 . . . . . (2)
+
+
+Those space-time points (events) which satisfy (1) must also satisfy
+(2). Obviously this will be the case when the relation
+
+(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).
+
+
+is fulfilled in general, where λ indicates a constant; for, according
+to (3), the disappearance of (_x_ – _ct_) involves the disappearance of
+(_x′_ – _ct′_).
+
+If we apply quite similar considerations to light rays which are being
+transmitted along the negative _x_-axis, we obtain the condition
+
+(_x′_ + _ct′_) = (_x + ct_) . . . (4).
+
+
+By adding (or subtracting) equations (3) and (4), and introducing for
+convenience the constants _a_ and _b_ in place of the constants λ and μ
+where
+
+image038
+
+
+and
+
+image039
+
+
+we obtain the equations
+
+image040
+
+
+We should thus have the solution of our problem, if the constants _a_
+and _b_ were known. These result from the following discussion.
+
+For the origin of _K′_ we have permanently _x′_ = 0, and hence
+according to the first of the equations (5)
+
+image041
+
+
+If we call _v_ the velocity with which the origin of _K′_ is moving
+relative to _K_, we then have
+
+image042
+
+
+The same value _v_ can be obtained from equations (5), if we calculate
+the velocity of another point of _K′_ relative to _K_, or the velocity
+(directed towards the negative _x_-axis) of a point of _K_ with respect
+to _K′_. In short, we can designate _v_ as the relative velocity of the
+two systems.
+
+Furthermore, the principle of relativity teaches us that, as judged
+from K, the length of a unit measuring-rod which is at rest with
+reference to _K′_ must be exactly the same as the length, as judged
+from _K′_, of a unit measuring-rod which is at rest relative to _K_. In
+order to see how the points of the _x′_-axis appear as viewed from _K_,
+we only require to take a “snapshot” of _K′_ from _K_; this means that
+we have to insert a particular value of _t_ (time of _K_), _e.g._ _t_ =
+0. For this value of _t_ we then obtain from the first of the equations
+(5)
+
+_x′_ = _ax_
+
+
+Two points of the _x′_-axis which are separated by the distance Δ_x′_ =
+1 when measured in the _K′_ system are thus separated in our
+instantaneous photograph by the distance
+
+image043
+
+
+But if the snapshot be taken from _K′_(_t′_ = 0), and if we eliminate
+_t_ from the equations (5), taking into account the expression (6), we
+obtain
+
+image044
+
+
+From this we conclude that two points on the _x_-axis separated by the
+distance 1 (relative to _K_) will be represented on our snapshot by the
+distance
+
+image045
+
+
+But from what has been said, the two snapshots must be identical; hence
+Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain
+
+image046
+
+
+The equations (6) and (7_b_) determine the constants _a_ and _b_. By
+inserting the values of these constants in (5), we obtain the first and
+the fourth of the equations given in Section XI.
+
+image047
+
+
+Thus we have obtained the Lorentz transformation for events on the
+_x_-axis. It satisfies the condition
+
+_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a).
+
+
+The extension of this result, to include events which take place
+outside the _x_-axis, is obtained by retaining equations (8) and
+supplementing them by the relations
+
+image048
+
+
+In this way we satisfy the postulate of the constancy of the velocity
+of light _in vacuo_ for rays of light of arbitrary direction, both for
+the system _K_ and for the system _K′_. This may be shown in the
+following manner.
+
+We suppose a light-signal sent out from the origin of _K_ at the time
+_t_ = 0. It will be propagated according to the equation
+
+image049
+
+
+or, if we square this equation, according to the equation
+
+_x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10).
+
+
+It is required by the law of propagation of light, in conjunction with
+the postulate of relativity, that the transmission of the signal in
+question should take place—as judged from _K′_—in accordance with the
+corresponding formula
+
+_r′_ = _ct′_
+
+
+or,
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).
+
+
+In order that equation (10_a_) may be a consequence of equation (10),
+we must have
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2)
+(11).
+
+
+Since equation (8_a_) must hold for points on the _x_-axis, we thus
+have σ = 1. It is easily seen that the Lorentz transformation really
+satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_)
+and (9), and hence also of (8) and (9). We have thus derived the
+Lorentz transformation.
+
+The Lorentz transformation represented by (8) and (9) still requires to
+be generalised. Obviously it is immaterial whether the axes of _K′_ be
+chosen so that they are spatially parallel to those of _K_. It is also
+not essential that the velocity of translation of _K′_ with respect to
+_K_ should be in the direction of the _x_-axis. A simple consideration
+shows that we are able to construct the Lorentz transformation in this
+general sense from two kinds of transformations, viz. from Lorentz
+transformations in the special sense and from purely spatial
+transformations. which corresponds to the replacement of the
+rectangular co-ordinate system by a new system with its axes pointing
+in other directions.
+
+Mathematically, we can characterise the generalised Lorentz
+transformation thus:
+
+It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions
+of _x, y, x, t_, of such a kind that the relation
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2
+(11_a_).
+
+
+is satisficd identically. That is to say: If we substitute their
+expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the
+left-hand side, then the left-hand side of (11_a_) agrees with the
+right-hand side.
+
+
+APPENDIX II
+
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)
+
+(SUPPLEMENTARY TO SECTION XVII)
+
+We can characterise the Lorentz transformation still more simply if we
+introduce the imaginary
+
+image031
+
+
+in place of _t_, as time-variable. If, in accordance with this, we
+insert
+
+image050
+
+
+and similarly for the accented system _K′_, then the condition which is
+identically satisfied by the transformation can be expressed thus:
+
+_x_1′2 + _x_2′2 + _x_3′2 + _x_4′2 = _x_12 + _x_22 + _x_32 + _x_42 (12).
+
+
+That is, by the afore-mentioned choice of “coordinates,” (11_a_) [see
+the end of Appendix II] is transformed into this equation.
+
+We see from (12) that the imaginary time co-ordinate _x_4, enters into
+the condition of transformation in exactly the same way as the space
+co-ordinates _x_1, _x_2, _x_3. It is due to this fact that, according
+to the theory of relativity, the “time” _x_4, enters into natural laws
+in the same form as the space co ordinates _x_1, _x_2, _x_3.
+
+A four-dimensional continuum described by the “co-ordinates” _x_1,
+_x_2, _x_3, _x_4, was called “world” by Minkowski, who also termed a
+point-event a “world-point.” From a “happening” in three-dimensional
+space, physics becomes, as it were, an “existence” in the
+four-dimensional “world.”
+
+This four-dimensional “world” bears a close similarity to the
+three-dimensional “space” of (Euclidean) analytical geometry. If we
+introduce into the latter a new Cartesian co-ordinate system (_x′_1,
+_x′_2, _x′_3) with the same origin, then _x′_1, _x′_2, _x′_3, are
+linear homogeneous functions of _x_1, _x_2, _x_3 which identically
+satisfy the equation
+
+_x_1′2 + _x_2′2 + _x_3′2 = _x_12 + _x_22 + _x_32
+
+
+The analogy with (12) is a complete one. We can regard Minkowski’s
+“world” in a formal manner as a four-dimensional Euclidean space (with
+an imaginary time coordinate); the Lorentz transformation corresponds
+to a “rotation” of the co-ordinate system in the four-dimensional
+“world.”
+
+
+APPENDIX III
+
+THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
+
+From a systematic theoretical point of view, we may imagine the process
+of evolution of an empirical science to be a continuous process of
+induction. Theories are evolved and are expressed in short compass as
+statements of a large number of individual observations in the form of
+empirical laws, from which the general laws can be ascertained by
+comparison. Regarded in this way, the development of a science bears
+some resemblance to the compilation of a classified catalogue. It is,
+as it were, a purely empirical enterprise.
+
+But this point of view by no means embraces the whole of the actual
+process; for it slurs over the important part played by intuition and
+deductive thought in the development of an exact science. As soon as a
+science has emerged from its initial stages, theoretical advances are
+no longer achieved merely by a process of arrangement. Guided by
+empirical data, the investigator rather develops a system of thought
+which, in general, is built up logically from a small number of
+fundamental assumptions, the so-called axioms. We call such a system of
+thought a _theory_. The theory finds the justification for its
+existence in the fact that it correlates a large number of single
+observations, and it is just here that the “truth” of the theory lies.
+
+Corresponding to the same complex of empirical data, there may be
+several theories, which differ from one another to a considerable
+extent. But as regards the deductions from the theories which are
+capable of being tested, the agreement between the theories may be so
+complete that it becomes difficult to find any deductions in which the
+two theories differ from each other. As an example, a case of general
+interest is available in the province of biology, in the Darwinian
+theory of the development of species by selection in the struggle for
+existence, and in the theory of development which is based on the
+hypothesis of the hereditary transmission of acquired characters.
+
+We have another instance of far-reaching agreement between the
+deductions from two theories in Newtonian mechanics on the one hand,
+and the general theory of relativity on the other. This agreement goes
+so far, that up to the present we have been able to find only a few
+deductions from the general theory of relativity which are capable of
+investigation, and to which the physics of pre-relativity days does not
+also lead, and this despite the profound difference in the fundamental
+assumptions of the two theories. In what follows, we shall again
+consider these important deductions, and we shall also discuss the
+empirical evidence appertaining to them which has hitherto been
+obtained.
+
+(_a_) Motion of the Perihelion of Mercury
+
+According to Newtonian mechanics and Newton’s law of gravitation, a
+planet which is revolving round the sun would describe an ellipse round
+the latter, or, more correctly, round the common centre of gravity of
+the sun and the planet. In such a system, the sun, or the common centre
+of gravity, lies in one of the foci of the orbital ellipse in such a
+manner that, in the course of a planet-year, the distance sun-planet
+grows from a minimum to a maximum, and then decreases again to a
+minimum. If instead of Newton’s law we insert a somewhat different law
+of attraction into the calculation, we find that, according to this new
+law, the motion would still take place in such a manner that the
+distance sun-planet exhibits periodic variations; but in this case the
+angle described by the line joining sun and planet during such a period
+(from perihelion—closest proximity to the sun—to perihelion) would
+differ from 360°. The line of the orbit would not then be a closed one
+but in the course of time it would fill up an annular part of the
+orbital plane, viz. between the circle of least and the circle of
+greatest distance of the planet from the sun.
+
+According also to the general theory of relativity, which differs of
+course from the theory of Newton, a small variation from the
+Newton-Kepler motion of a planet in its orbit should take place, and in
+such away, that the angle described by the radius sun-planet between
+one perhelion and the next should exceed that corresponding to one
+complete revolution by an amount given by
+
+image051
+
+
+(_N.B._—One complete revolution corresponds to the angle 2π in the
+absolute angular measure customary in physics, and the above expression
+given the amount by which the radius sun-planet exceeds this angle
+during the interval between one perihelion and the next.) In this
+expression _a_ represents the major semi-axis of the ellipse, _e_ its
+eccentricity, _c_ the velocity of light, and _T_ the period of
+revolution of the planet. Our result may also be stated as follows:
+According to the general theory of relativity, the major axis of the
+ellipse rotates round the sun in the same sense as the orbital motion
+of the planet. Theory requires that this rotation should amount to 43
+seconds of arc per century for the planet Mercury, but for the other
+Planets of our solar system its magnitude should be so small that it
+would necessarily escape detection.[26]
+
+
+ [26] Especially since the next planet Venus has an orbit that is
+ almost an exact circle, which makes it more difficult to locate the
+ perihelion with precision.
+
+
+In point of fact, astronomers have found that the theory of Newton does
+not suffice to calculate the observed motion of Mercury with an
+exactness corresponding to that of the delicacy of observation
+attainable at the present time. After taking account of all the
+disturbing influences exerted on Mercury by the remaining planets, it
+was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
+perihelial movement of the orbit of Mercury remained over, the amount
+of which does not differ sensibly from the above mentioned +43 seconds
+of arc per century. The uncertainty of the empirical result amounts to
+a few seconds only.
+
+(_b_) Deflection of Light by a Gravitational Field
+
+image052
+
+
+In Section XXII it has been already mentioned that according to the
+general theory of relativity, a ray of light will experience a
+curvature of its path when passing through a gravitational field, this
+curvature being similar to that experienced by the path of a body which
+is projected through a gravitational field. As a result of this theory,
+we should expect that a ray of light which is passing close to a
+heavenly body would be deviated towards the latter. For a ray of light
+which passes the sun at a distance of Δ sun-radii from its centre, the
+angle of deflection (α) should amount to
+
+image053
+
+
+It may be added that, according to the theory, half of this deflection
+is produced by the Newtonian field of attraction of the sun, and the
+other half by the geometrical modification (“curvature”) of space
+caused by the sun.
+
+This result admits of an experimental test by means of the photographic
+registration of stars during a total eclipse of the sun. The only
+reason why we must wait for a total eclipse is because at every other
+time the atmosphere is so strongly illuminated by the light from the
+sun that the stars situated near the sun’s disc are invisible. The
+predicted effect can be seen clearly from the accompanying diagram. If
+the sun (_S_) were not present, a star which is practically infinitely
+distant would be seen in the direction _D_1, as observed front the
+earth. But as a consequence of the deflection of light from the star by
+the sun, the star will be seen in the direction _D_2, _i.e._ at a
+somewhat greater distance from the centre of the sun than corresponds
+to its real position.
+
+In practice, the question is tested in the following way. The stars in
+the neighbourhood of the sun are photographed during a solar eclipse.
+
+In addition, a second photograph of the same stars is taken when the
+sun is situated at another position in the sky, _i.e._ a few months
+earlier or later. As compared with the standard photograph, the
+positions of the stars on the eclipse-photograph ought to appear
+displaced radially outwards (away from the centre of the sun) by an
+amount corresponding to the angle _a_.
+
+We are indebted to the [British] Royal Society and to the Royal
+Astronomical Society for the investigation of this important deduction.
+Undaunted by the [first world] war and by difficulties of both a
+material and a psychological nature aroused by the war, these societies
+equipped two expeditions—to Sobral (Brazil), and to the island of
+Principe (West Africa)—and sent several of Britain’s most celebrated
+astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to
+obtain photographs of the solar eclipse of 29th May, 1919. The relative
+discrepancies to be expected between the stellar photographs obtained
+during the eclipse and the comparison photographs amounted to a few
+hundredths of a millimetre only. Thus great accuracy was necessary in
+making the adjustments required for the taking of the photographs, and
+in their subsequent measurement.
+
+The results of the measurements confirmed the theory in a thoroughly
+satisfactory manner. The rectangular components of the observed and of
+the calculated deviations of the stars (in seconds of arc) are set
+forth in the following table of results:
+
+image054
+
+
+(_c_) Displacement of Spectral Lines Towards the Red
+
+In Section XXIII it has been shown that in a system _K′_ which is in
+rotation with regard to a Galileian system _K_, clocks of identical
+construction, and which are considered at rest with respect to the
+rotating reference-body, go at rates which are dependent on the
+positions of the clocks. We shall now examine this dependence
+quantitatively. A clock, which is situated at a distance r from the
+centre of the disc, has a velocity relative to _K_ which is given by
+
+_v_ = ω_r_,
+
+
+where ω represents the angular velocity of rotation of the disc _K′_
+with respect to _K_. If _v_0, represents the number of ticks of the
+clock per unit time (“rate” of the clock) relative to _K_ when the
+clock is at rest, then the “rate” of the clock (_v_) when it is moving
+relative to _K_ with a velocity _v_, but at rest with respect to the
+disc, will, in accordance with Section XII, be given by
+
+image055
+
+
+or with sufficient accuracy by
+
+image056
+
+
+This expression may also be stated in the following form:
+
+image057
+
+
+If we represent the difference of potential of the centrifugal force
+between the position of the clock and the centre of the disc by φ,
+_i.e._ the work, considered negatively, which must be performed on the
+unit of mass against the centrifugal force in order to transport it
+from the position of the clock on the rotating disc to the centre of
+the disc, then we have
+
+image058
+
+
+From this it follows that
+
+image059
+
+
+In the first place, we see from this expression that two clocks of
+identical construction will go at different rates when situated at
+different distances from the centre of the disc. This result is also
+valid from the standpoint of an observer who is rotating with the disc.
+
+Now, as judged from the disc, the latter is in a gravitational field of
+potential φ, hence the result we have obtained will hold quite
+generally for gravitational fields. Furthermore, we can regard an atom
+which is emitting spectral lines as a clock, so that the following
+statement will hold:
+
+_An atom absorbs or emits light of a frequency which is dependent on
+the potential of the gravitational field in which it is situated._
+
+The frequency of an atom situated on the surface of a heavenly body
+will be somewhat less than the frequency of an atom of the same element
+which is situated in free space (or on the surface of a smaller
+celestial body).
+
+Now φ = – _K (M/r)_, where _K_ is Newton’s constant of gravitation, and
+_M_ is the mass of the heavenly body. Thus a displacement towards the
+red ought to take place for spectral lines produced at the surface of
+stars as compared with the spectral lines of the same element produced
+at the surface of the earth, the amount of this displacement being
+
+image060
+
+
+For the sun, the displacement towards the red predicted by theory
+amounts to about two millionths of the wave-length. A trustworthy
+calculation is not possible in the case of the stars, because in
+general neither the mass _M_ nor the radius _r_ are known.
+
+It is an open question whether or not this effect exists, and at the
+present time (1920) astronomers are working with great zeal towards the
+solution. Owing to the smallness of the effect in the case of the sun,
+it is difficult to form an opinion as to its existence. Whereas Grebe
+and Bachem (Bonn), as a result of their own measurements and those of
+Evershed and Schwarzschild on the cyanogen bands, have placed the
+existence of the effect almost beyond doubt, while other investigators,
+particularly St. John, have been led to the opposite opinion in
+consequence of their measurements.
+
+Mean displacements of lines towards the less refrangible end of the
+spectrum are certainly revealed by statistical investigations of the
+fixed stars; but up to the present the examination of the available
+data does not allow of any definite decision being arrived at, as to
+whether or not these displacements are to be referred in reality to the
+effect of gravitation. The results of observation have been collected
+together, and discussed in detail from the standpoint of the question
+which has been engaging our attention here, in a paper by E. Freundlich
+entitled “Zur Prüfung der allgemeinen Relativitäts-Theorie” (_Die
+Naturwissenschaften_, 1919, No. 35, p. 520: Julius Springer, Berlin).
+
+At all events, a definite decision will be reached during the next few
+years. If the displacement of spectral lines towards the red by the
+gravitational potential does not exist, then the general theory of
+relativity will be untenable. On the other hand, if the cause of the
+displacement of spectral lines be definitely traced to the
+gravitational potential, then the study of this displacement will
+furnish us with important information as to the mass of the heavenly
+bodies.[27]
+
+
+ [27] The displacement of spectral lines towards the red end of the
+ spectrum was definitely established by Adams in 1924, by observations
+ on the dense companion of Sirius, for which the effect is about thirty
+ times greater than for the Sun. R.W.L.—translator
+
+
+APPENDIX IV
+
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
+
+(SUPPLEMENTARY TO SECTION XXXII)
+
+Since the publication of the first edition of this little book, our
+knowledge about the structure of space in the large (“cosmological
+problem”) has had an important development, which ought to be mentioned
+even in a popular presentation of the subject.
+
+My original considerations on the subject were based on two hypotheses:
+
+(1) There exists an average density of matter in the whole of space
+which is everywhere the same and different from zero.
+
+(2) The magnitude (“radius”) of space is independent of time.
+
+Both these hypotheses proved to be consistent, according to the general
+theory of relativity, but only after a hypothetical term was added to
+the field equations, a term which was not required by the theory as
+such nor did it seem natural from a theoretical point of view
+(“cosmological term of the field equations”).
+
+Hypothesis (2) appeared unavoidable to me at the time, since I thought
+that one would get into bottomless speculations if one departed from
+it.
+
+However, already in the ’twenties, the Russian mathematician Friedman
+showed that a different hypothesis was natural from a purely
+theoretical point of view. He realized that it was possible to preserve
+hypothesis (1) without introducing the less natural cosmological term
+into the field equations of gravitation, if one was ready to drop
+hypothesis (2). Namely, the original field equations admit a solution
+in which the “world radius” depends on time (expanding space). In that
+sense one can say, according to Friedman, that the theory demands an
+expansion of space.
+
+A few years later Hubble showed, by a special investigation of the
+extra-galactic nebulae (“milky ways”), that the spectral lines emitted
+showed a red shift which increased regularly with the distance of the
+nebulae. This can be interpreted in regard to our present knowledge
+only in the sense of Doppler’s principle, as an expansive motion of the
+system of stars in the large—as required, according to Friedman, by the
+field equations of gravitation. Hubble’s discovery can, therefore, be
+considered to some extent as a confirmation of the theory.
+
+There does arise, however, a strange difficulty. The interpretation of
+the galactic line-shift discovered by Hubble as an expansion (which can
+hardly be doubted from a theoretical point of view), leads to an origin
+of this expansion which lies “only” about 109 years ago, while physical
+astronomy makes it appear likely that the development of individual
+stars and systems of stars takes considerably longer. It is in no way
+known how this incongruity is to be overcome.
+
+I further want to remark that the theory of expanding space, together
+with the empirical data of astronomy, permit no decision to be reached
+about the finite or infinite character of (three-dimensional) space,
+while the original “static” hypothesis of space yielded the closure
+(finiteness) of space.
+
+_K_ = co-ordinate system
+
+_x, y_ = two-dimensional co-ordinates
+
+_x, y, z_ = three-dimensional co-ordinates
+
+_x, y, z, t_ = four-dimensional co-ordinates
+
+_t_ = time
+
+_I_ = distance
+
+_v_ = velocity
+
+_F_ = force
+
+_G_ = gravitational field
+
+
+
+
+*** END OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***
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+<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of Relativity: The Special and General Theory, by Albert Einstein</div>
+<div style='display:block; margin:1em 0'>
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+of the Project Gutenberg License included with this eBook or online
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+country where you are located before using this eBook.
+</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: Relativity: The Special and General Theory</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: Albert Einstein</div>
+<div style='display:block; margin:1em 0'>Release Date: October 1, 2009 [eBook #30155]<br />
+[Most recently updated: May 2, 2023]</div>
+<div style='display:block; margin:1em 0'>Language: English</div>
+<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Robert Hux</div>
+<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***</div>
+
+<div class="fig" style="width:70%;">
+<img src="images/cover.jpg" style="width:100%;" alt="cover " /><br/><br/>
+</div>
+
+<h1>Relativity: The Special and General Theory</h1>
+
+<h2>by Albert Einstein</h2>
+
+<h4>Authorised Translation by Robert W. Lawson</h4>
+<hr />
+
+<p>ALBERT EINSTEIN REFERENCE ARCHIVE</p>
+
+<p>RELATIVITY: THE SPECIAL AND GENERAL THEORY</p>
+
+<p>BY ALBERT EINSTEIN<br/><br/></p>
+
+<p>Written: 1916 (this revised edition: 1924)</p>
+
+<p>Source: Relativity: The Special and General Theory (1920)</p>
+
+<p>Publisher: Methuen &amp; Co Ltd</p>
+
+<p>First Published: December, 1916</p>
+
+<p>Translated: Robert W. Lawson (Authorised translation)</p>
+
+<p>Transcription/Markup: Brian Basgen</p>
+
+<p>Transcription to text: Gregory B. Newby</p>
+
+<p>Thanks to: Einstein Reference Archive (marxists.org)</p>
+
+<p>The Einstein Reference Archive is online at:</p>
+
+<p>http://www.marxists.org/reference/archive/einstein/index.htm<br/><br/></p>
+
+<h3>Contents</h3>
+
+<table summary="">
+
+<tr>
+<td> <a href="#pref01">Preface</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part01"><b>Part I: The Special Theory of Relativity</b></a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap01">I. Physical Meaning of Geometrical Propositions</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap02">II. The System of Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap03">III. Space and Time in Classical Mechanics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap04">IV. The Galileian System of Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap05">V. The Principle of Relativity (in the Restricted Sense)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap06">VI. The Theorem of the Addition of Velocities employed in Classical Mechanics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap07">VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap08">VIII. On the Idea of Time in Physics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap09">IX. The Relativity of Simultaneity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap10">X. On the Relativity of the Conception of Distance</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap11">XI. The Lorentz Transformation</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap12">XII. The Behaviour of Measuring-Rods and Clocks in Motion</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap13">XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap14">XIV. The Heuristic Value of the Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap15">XV. General Results of the Theory</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap16">XVI. Experience and the Special Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap17">XVII. Minkowski’s Four-dimensional Space</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part02"><b>Part II: The General Theory of Relativity</b></a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap18">XVIII. Special and General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap19">XIX. The Gravitational Field</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap20">XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap21">XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap22">XXII. A Few Inferences from the General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap23">XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap24">XXIV. Euclidean and non-Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap25">XXV. Gaussian Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap26">XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap27">XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap28">XXVIII. Exact Formulation of the General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap29">XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part03"><b>Part III: Considerations on the Universe as a Whole</b></a></td>
+</tr>
+<tr>
+<td> <a href="#chap30">XXX. Cosmological Difficulties of Newton’s Theory</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap31">XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap32">XXXII. The Structure of Space According to the General Theory of Relativity</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#chap33">Appendices:</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap34">I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap35">II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap36">III. The Experimental Confirmation of the General Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap37">IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap38">V. Relativity and the Problem of Space</a></td>
+</tr>
+
+</table>
+
+<p>
+Note: The fifth Appendix was added by Einstein at the time of the fifteenth
+re-printing of this book; and as a result is still under copyright restrictions
+so cannot be added without the permission of the publisher.
+</p>
+
+<div class="chapter">
+
+<h3><a name="pref01"></a>PREFACE</h3>
+
+<p>
+The present book is intended, as far as possible, to give an exact insight into
+the theory of Relativity to those readers who, from a general scientific and
+philosophical point of view, are interested in the theory, but who are not
+conversant with the mathematical apparatus of theoretical physics. The work
+presumes a standard of education corresponding to that of a university
+matriculation examination, and, despite the shortness of the book, a fair
+amount of patience and force of will on the part of the reader. The author has
+spared himself no pains in his endeavour to present the main ideas in the
+simplest and most intelligible form, and on the whole, in the sequence and
+connection in which they actually originated. In the interest of clearness, it
+appeared to me inevitable that I should repeat myself frequently, without
+paying the slightest attention to the elegance of the presentation. I adhered
+scrupulously to the precept of that brilliant theoretical physicist L.
+Boltzmann, according to whom matters of elegance ought to be left to the tailor
+and to the cobbler. I make no pretence of having withheld from the reader
+difficulties which are inherent to the subject. On the other hand, I have
+purposely treated the empirical physical foundations of the theory in a
+“step-motherly” fashion, so that readers unfamiliar with physics may
+not feel like the wanderer who was unable to see the forest for the trees. May
+the book bring some one a few happy hours of suggestive thought!
+</p>
+
+<p>
+December, 1916
+</p>
+
+<p class="right"> A. EINSTEIN
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part01"></a>PART I: THE SPECIAL THEORY OF RELATIVITY</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap01"></a>I.<br/>
+PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS</h3>
+
+<p>
+In your schooldays most of you who read this book made acquaintance with the
+noble building of Euclid’s geometry, and you remember—perhaps with more
+respect than love—the magnificent structure, on the lofty staircase of which
+you were chased about for uncounted hours by conscientious teachers. By reason
+of our past experience, you would certainly regard everyone with disdain who
+should pronounce even the most out-of-the-way proposition of this science to be
+untrue. But perhaps this feeling of proud certainty would leave you immediately
+if some one were to ask you: “What, then, do you mean by the assertion
+that these propositions are true?” Let us proceed to give this question a
+little consideration.
+</p>
+
+<p>
+Geometry sets out from certain conceptions such as “plane,”
+“point,” and “straight line,” with which we are able to
+associate more or less definite ideas, and from certain simple propositions
+(axioms) which, in virtue of these ideas, we are inclined to accept as
+“true.” Then, on the basis of a logical process, the justification of
+which we feel ourselves compelled to admit, all remaining propositions are
+shown to follow from those axioms, <i>i.e.</i> they are proven. A proposition is then
+correct (“true”) when it has been derived in the recognised manner
+from the axioms. The question of “truth” of the individual
+geometrical propositions is thus reduced to one of the “truth” of the
+axioms. Now it has long been known that the last question is not only
+unanswerable by the methods of geometry, but that it is in itself entirely
+without meaning. We cannot ask whether it is true that only one straight line
+goes through two points. We can only say that Euclidean geometry deals with
+things called “straight lines,” to each of which is ascribed the
+property of being uniquely determined by two points situated on it. The concept
+“true” does not tally with the assertions of pure geometry, because
+by the word “true” we are eventually in the habit of designating
+always the correspondence with a “real” object; geometry, however, is
+not concerned with the relation of the ideas involved in it to objects of
+experience, but only with the logical connection of these ideas among
+themselves.
+</p>
+
+<p>
+It is not difficult to understand why, in spite of this, we feel constrained to
+call the propositions of geometry “true.” Geometrical ideas
+correspond to more or less exact objects in nature, and these last are
+undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought
+to refrain from such a course, in order to give to its structure the largest
+possible logical unity. The practice, for example, of seeing in a
+“distance” two marked positions on a practically rigid body is
+something which is lodged deeply in our habit of thought. We are accustomed
+further to regard three points as being situated on a straight line, if their
+apparent positions can be made to coincide for observation with one eye, under
+suitable choice of our place of observation.
+</p>
+
+<p>
+If, in pursuance of our habit of thought, we now supplement the propositions of
+Euclidean geometry by the single proposition that two points on a practically
+rigid body always correspond to the same distance (line-interval),
+independently of any changes in position to which we may subject the body, the
+propositions of Euclidean geometry then resolve themselves into propositions on
+the possible relative position of practically rigid bodies.<a
+href="#linknote-1" name="linknoteref-1" id="linknoteref-1">[1]</a> Geometry
+which has been supplemented in this way is then to be treated as a branch of
+physics. We can now legitimately ask as to the “truth” of
+geometrical propositions interpreted in this way, since we are justified in
+asking whether these propositions are satisfied for those real things we have
+associated with the geometrical ideas. In less exact terms we can express this
+by saying that by the “truth” of a geometrical proposition in this
+sense we understand its validity for a construction with rule and compasses.
+</p>
+
+<p>
+<a name="linknote-1" id="linknote-1">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-1"> [1]</a><br/> It follows that a natural object is
+associated also with a straight line. Three points <i>A, B</i> and <i>C</i> on a rigid body
+thus lie in a straight line when the points <i>A</i> and <i>C</i> being given, <i>B</i> is chosen
+such that the sum of the distances <i>AB</i> and <i>BC</i> is as short as possible. This
+incomplete suggestion will suffice for the present purpose.
+</p>
+
+<p>
+Of course the conviction of the “truth” of geometrical propositions
+in this sense is founded exclusively on rather incomplete experience. For the
+present we shall assume the “truth” of the geometrical propositions,
+then at a later stage (in the general theory of relativity) we shall see that
+this “truth” is limited, and we shall consider the extent of its
+limitation.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap02"></a>II.<br/>
+THE SYSTEM OF CO-ORDINATES</h3>
+
+<p>
+On the basis of the physical interpretation of distance which has been
+indicated, we are also in a position to establish the distance between two
+points on a rigid body by means of measurements. For this purpose we require a
+“distance” (rod <i>S</i>) which is to be used once and for all, and
+which we employ as a standard measure. If, now, <i>A</i> and <i>B</i> are two
+points on a rigid body, we can construct the line joining them according to the
+rules of geometry; then, starting from <i>A</i>, we can mark off the distance
+<i>S</i> time after time until we reach <i>B</i>. The number of these
+operations required is the numerical measure of the distance <i>AB</i>. This is
+the basis of all measurement of length.<a href="#linknote-2"
+name="linknoteref-2" id="linknoteref-2">[2]</a>
+</p>
+
+<p>
+<a name="linknote-2" id="linknote-2">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-2"> [2]</a><br/> Here we have assumed that there is
+nothing left over <i>i.e.</i> that the measurement gives a whole number. This
+difficulty is got over by the use of divided measuring-rods, the introduction
+of which does not demand any fundamentally new method.
+</p>
+
+
+<p>
+Every description of the scene of an event or of the position of an object in
+space is based on the specification of the point on a rigid body (body of
+reference) with which that event or object coincides. This applies not only to
+scientific description, but also to everyday life. If I analyse the place
+specification “Trafalgar Square, London”<a href="#linknote-3" name="linknoteref-3" id="linknoteref-3">[3]</a> I arrive at
+the following result. The earth is the rigid body to which the specification of
+place refers; “Trafalgar Square, London” is a well-defined point, to
+which a name has been assigned, and with which the event coincides in
+space.<a href="#linknote-4" name="linknoteref-4" id="linknoteref-4">[4]</a>
+</p>
+
+<p>
+<a name="linknote-3" id="linknote-3">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-3"> [3]</a><br/> 
+I have chosen this as being more familiar to the English reader than the
+“Potzdammer Platz, Berlin,” which is referred to in the original.
+(R. W. L.)
+</p>
+
+<p>
+<a name="linknote-4" id="linknote-4">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-4"> [4]</a><br/> It is not necessary here to investigate
+further the significance of the expression “coincidence in space.”
+This conception is sufficiently obvious to ensure that differences of opinion
+are scarcely likely to arise as to its applicability in practice.
+</p>
+
+<p>
+This primitive method of place specification deals only with places on the
+surface of rigid bodies, and is dependent on the existence of points on this
+surface which are distinguishable from each other. But we can free ourselves
+from both of these limitations without altering the nature of our specification
+of position. If, for instance, a cloud is hovering over Trafalgar Square, then we
+can determine its position relative to the surface of the earth by erecting a
+pole perpendicularly on the Square, so that it reaches the cloud. The length of
+the pole measured with the standard measuring-rod, combined with the
+specification of the position of the foot of the pole, supplies us with a
+complete place specification. On the basis of this illustration, we are able to
+see the manner in which a refinement of the conception of position has been
+developed.
+</p>
+
+<p>
+(<i>a</i>) We imagine the rigid body, to which the place specification is referred,
+supplemented in such a manner that the object whose position we require is
+reached by the completed rigid body.
+</p>
+
+<p>
+(<i>b</i>) In locating the position of the object, we make use of a number (here the
+length of the pole measured with the measuring-rod) instead of designated
+points of reference.
+</p>
+
+<p>
+(<i>c</i>) We speak of the height of the cloud even when the pole which reaches the
+cloud has not been erected. By means of optical observations of the cloud from
+different positions on the ground, and taking into account the properties of
+the propagation of light, we determine the length of the pole we should have
+required in order to reach the cloud.
+</p>
+
+<p>
+From this consideration we see that it will be advantageous if, in the
+description of position, it should be possible by means of numerical measures
+to make ourselves independent of the existence of marked positions (possessing
+names) on the rigid body of reference. In the physics of measurement this is
+attained by the application of the Cartesian system of co-ordinates.
+</p>
+
+<p>
+This consists of three plane surfaces perpendicular to each other and rigidly
+attached to a rigid body. Referred to a system of co-ordinates, the scene of
+any event will be determined (for the main part) by the specification of the
+lengths of the three perpendiculars or co-ordinates (<i>x, y, z</i>) which can be
+dropped from the scene of the event to those three plane surfaces. The lengths
+of these three perpendiculars can be determined by a series of manipulations
+with rigid measuring-rods performed according to the rules and methods laid
+down by Euclidean geometry.
+</p>
+
+<p>
+In practice, the rigid surfaces which constitute the system of co-ordinates are
+generally not available; furthermore, the magnitudes of the co-ordinates are
+not actually determined by constructions with rigid rods, but by indirect
+means. If the results of physics and astronomy are to maintain their clearness,
+the physical meaning of specifications of position must always be sought in
+accordance with the above considerations.<a href="#linknote-5" name="linknoteref-5" id="linknoteref-5">[5]</a>
+</p>
+
+<p>
+<a name="linknote-5" id="linknote-5">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-5"> [5]</a><br/> A refinement and modification of these
+views does not become necessary until we come to deal with the general theory
+of relativity, treated in the second part of this book.
+</p>
+
+<p>
+We thus obtain the following result: Every description of events in space
+involves the use of a rigid body to which such events have to be referred. The
+resulting relationship takes for granted that the laws of Euclidean geometry
+hold for “distances;” the “distance” being represented
+physically by means of the convention of two marks on a rigid body.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap03"></a>III.<br/>
+SPACE AND TIME IN CLASSICAL MECHANICS</h3>
+
+<p>
+The purpose of mechanics is to describe how bodies change their position in
+space with “time.” I should load my conscience with grave sins
+against the sacred spirit of lucidity were I to formulate the aims of mechanics
+in this way, without serious reflection and detailed explanations. Let us
+proceed to disclose these sins.
+</p>
+
+<p>
+It is not clear what is to be understood here by “position” and
+“space.” I stand at the window of a railway carriage which is
+travelling uniformly, and drop a stone on the embankment, without throwing it.
+Then, disregarding the influence of the air resistance, I see the stone descend
+in a straight line. A pedestrian who observes the misdeed from the footpath
+notices that the stone falls to earth in a parabolic curve. I now ask: Do the
+“positions” traversed by the stone lie “in reality” on a
+straight line or on a parabola? Moreover, what is meant here by motion “in
+space”? From the considerations of the previous section the answer is
+self-evident. In the first place we entirely shun the vague word
+“space,” of which, we must honestly acknowledge, we cannot form the
+slightest conception, and we replace it by “motion relative to a
+practically rigid body of reference.” The positions relative to the body
+of reference (railway carriage or embankment) have already been defined in
+detail in the preceding section. If instead of “body of reference” we
+insert “system of co-ordinates,” which is a useful idea for
+mathematical description, we are in a position to say: The stone traverses a
+straight line relative to a system of co-ordinates rigidly attached to the
+carriage, but relative to a system of co-ordinates rigidly attached to the
+ground (embankment) it describes a parabola. With the aid of this example it is
+clearly seen that there is no such thing as an independently existing
+trajectory (lit. “path-curve”<a href="#linknote-6" name="linknoteref-6" id="linknoteref-6">[6]</a>, but only a trajectory
+relative to a particular body of reference.
+</p>
+
+<p>
+<a name="linknote-6" id="linknote-6">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-6"> [6]</a><br/> That is, a curve along which the body
+moves.
+</p>
+
+<p>
+In order to have a <i>complete</i> description of the motion, we must specify how the
+body alters its position <i>with time; i.e.</i> for every point on the trajectory it
+must be stated at what time the body is situated there. These data must be
+supplemented by such a definition of time that, in virtue of this definition,
+these time-values can be regarded essentially as magnitudes (results of
+measurements) capable of observation. If we take our stand on the ground of
+classical mechanics, we can satisfy this requirement for our illustration in
+the following manner. We imagine two clocks of identical construction; the man
+at the railway-carriage window is holding one of them, and the man on the
+footpath the other. Each of the observers determines the position on his own
+reference-body occupied by the stone at each tick of the clock he is holding in
+his hand. In this connection we have not taken account of the inaccuracy
+involved by the finiteness of the velocity of propagation of light. With this
+and with a second difficulty prevailing here we shall have to deal in detail
+later.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap04"></a>IV.<br/>THE GALILEIAN SYSTEM OF CO-ORDINATES</h3>
+
+
+<p>
+As is well known, the fundamental law of the mechanics of Galilei-Newton, which
+is known as the <i>law of inertia</i>, can be stated thus: A body removed sufficiently
+far from other bodies continues in a state of rest or of uniform motion in a
+straight line. This law not only says something about the motion of the bodies,
+but it also indicates the reference-bodies or systems of coordinates,
+permissible in mechanics, which can be used in mechanical description. The
+visible fixed stars are bodies for which the law of inertia certainly holds to
+a high degree of approximation. Now if we use a system of co-ordinates which is
+rigidly attached to the earth, then, relative to this system, every fixed star
+describes a circle of immense radius in the course of an astronomical day, a
+result which is opposed to the statement of the law of inertia. So that if we
+adhere to this law we must refer these motions only to systems of coordinates
+relative to which the fixed stars do not move in a circle. A system of
+co-ordinates of which the state of motion is such that the law of inertia holds
+relative to it is called a “Galileian system of co-ordinates.” The
+laws of the mechanics of Galilei-Newton can be regarded as valid only for a
+Galileian system of co-ordinates.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap05"></a>V.<br/>
+THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)</h3>
+
+<p>
+In order to attain the greatest possible clearness, let us return to our
+example of the railway carriage supposed to be travelling uniformly. We call
+its motion a uniform translation (“uniform” because it is of constant
+velocity and direction, “translation” because although the carriage
+changes its position relative to the embankment yet it does not rotate in so
+doing). Let us imagine a raven flying through the air in such a manner that its
+motion, as observed from the embankment, is uniform and in a straight line. If
+we were to observe the flying raven from the moving railway carriage. we should
+find that the motion of the raven would be one of different velocity and
+direction, but that it would still be uniform and in a straight line. Expressed
+in an abstract manner we may say: If a mass <i>m</i> is moving uniformly in a
+straight line with respect to a co-ordinate system <i>K</i>, then it will also be
+moving uniformly and in a straight line relative to a second co-ordinate system
+<i>K&#x2032;</i> provided that the latter is executing a uniform translatory motion with
+respect to <i>K</i>. In accordance with the discussion contained in the preceding
+section, it follows that:
+</p>
+
+<p>
+If <i>K</i> is a Galileian co-ordinate system. then every other co-ordinate system <i>K&#x2032;</i>
+is a Galileian one, when, in relation to <i>K</i>, it is in a condition of uniform
+motion of translation. Relative to <i>K&#x2032;</i> the mechanical laws of Galilei-Newton
+hold good exactly as they do with respect to <i>K</i>.
+</p>
+
+<p>
+We advance a step farther in our generalisation when we express the tenet thus:
+If, relative to <i>K</i>, <i>K&#x2032;</i> is a uniformly moving co-ordinate system devoid of
+rotation, then natural phenomena run their course with respect to <i>K&#x2032;</i> according
+to exactly the same general laws as with respect to <i>K</i>. This statement is called
+the <i>principle of relativity</i> (in the restricted sense).
+</p>
+
+<p>
+As long as one was convinced that all natural phenomena were capable of
+representation with the help of classical mechanics, there was no need to doubt
+the validity of this principle of relativity. But in view of the more recent
+development of electrodynamics and optics it became more and more evident that
+classical mechanics affords an insufficient foundation for the physical
+description of all natural phenomena. At this juncture the question of the
+validity of the principle of relativity became ripe for discussion, and it did
+not appear impossible that the answer to this question might be in the
+negative.
+</p>
+
+<p>
+Nevertheless, there are two general facts which at the outset speak very much
+in favour of the validity of the principle of relativity. Even though classical
+mechanics does not supply us with a sufficiently broad basis for the
+theoretical presentation of all physical phenomena, still we must grant it a
+considerable measure of “truth,” since it supplies us with the actual
+motions of the heavenly bodies with a delicacy of detail little short of
+wonderful. The principle of relativity must therefore apply with great accuracy
+in the domain of <i>mechanics</i>. But that a principle of such broad generality
+should hold with such exactness in one domain of phenomena, and yet should be
+invalid for another, is <i>a priori</i> not very probable.
+</p>
+
+<p>
+We now proceed to the second argument, to which, moreover, we shall return
+later. If the principle of relativity (in the restricted sense) does not hold,
+then the Galileian co-ordinate systems <i>K, K&#x2032;, K&#x2033;</i>, etc., which are moving
+uniformly relative to each other, will not be <i>equivalent</i> for the description of
+natural phenomena. In this case we should be constrained to believe that
+natural laws are capable of being formulated in a particularly simple manner,
+and of course only on condition that, from amongst all possible Galileian
+co-ordinate systems, we should have chosen <i>one</i> (<i>K<sub>0</sub></i>) of a particular
+state of motion as our body of reference. We should then be justified (because
+of its merits for the description of natural phenomena) in calling this system
+“absolutely at rest,” and all other Galileian systems <i>K</i>
+“in motion.” If, for instance, our embankment were the system
+<i>K<sub>0</sub></i> then our railway carriage would be a system <i>K</i>, relative to which
+less simple laws would hold than with respect to <i>K<sub>0</sub></i>. This diminished
+simplicity would be due to the fact that the carriage <i>K</i> would be in motion
+(<i>i.e.</i> “really”)with respect to <i>K<sub>0</sub></i>. In the general laws of
+nature which have been formulated with reference to <i>K</i>, the magnitude and
+direction of the velocity of the carriage would necessarily play a part. We
+should expect, for instance, that the note emitted by an organpipe placed with
+its axis parallel to the direction of travel would be different from that
+emitted if the axis of the pipe were placed perpendicular to this direction.
+</p>
+
+<p>
+Now in virtue of its motion in an orbit round the sun, our earth is comparable
+with a railway carriage travelling with a velocity of about 30 kilometres per
+second. If the principle of relativity were not valid we should therefore
+expect that the direction of motion of the earth at any moment would enter into
+the laws of nature, and also that physical systems in their behaviour would be
+dependent on the orientation in space with respect to the earth. For owing to
+the alteration in direction of the velocity of revolution of the earth in the
+course of a year, the earth cannot be at rest relative to the hypothetical
+system <i>K<sub>0</sub></i> throughout the whole year. However, the most careful
+observations have never revealed such anisotropic properties in terrestrial
+physical space, <i>i.e.</i> a physical non-equivalence of different directions. This
+is very powerful argument in favour of the principle of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap06"></a>VI.<br/>
+THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL MECHANICS</h3>
+
+<p>
+Let us suppose our old friend the railway carriage to be travelling along the
+rails with a constant velocity <i>v</i>, and that a man traverses the length of the
+carriage in the direction of travel with a velocity <i>w</i>. How quickly or, in other
+words, with what velocity <i>W</i> does the man advance relative to the embankment
+during the process? The only possible answer seems to result from the following
+consideration: If the man were to stand still for a second, he would advance
+relative to the embankment through a distance <i>v</i> equal numerically to the
+velocity of the carriage. As a consequence of his walking, however, he
+traverses an additional distance w relative to the carriage, and hence also
+relative to the embankment, in this second, the distance w being numerically
+equal to the velocity with which he is walking. Thus in total he covers the
+distance <i>W = v + w</i> relative to the embankment in the second considered. We shall
+see later that this result, which expresses the theorem of the addition of
+velocities employed in classical mechanics, cannot be maintained; in other
+words, the law that we have just written down does not hold in reality. For the
+time being, however, we shall assume its correctness.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap07"></a>VII.<br/>
+THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH THE
+PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+There is hardly a simpler law in physics than that according to which light is
+propagated in empty space. Every child at school knows, or believes he knows,
+that this propagation takes place in straight lines with a velocity <i>c</i> = 300,000
+km./sec. At all events we know with great exactness that this velocity is the
+same for all colours, because if this were not the case, the minimum of
+emission would not be observed simultaneously for different colours during the
+eclipse of a fixed star by its dark neighbour. By means of similar
+considerations based on observations of double stars, the Dutch astronomer De
+Sitter was also able to show that the velocity of propagation of light cannot
+depend on the velocity of motion of the body emitting the light. The assumption
+that this velocity of propagation is dependent on the direction “in
+space” is in itself improbable.
+</p>
+
+<p>
+In short, let us assume that the simple law of the constancy of the velocity of
+light <i>c</i> (in vacuum) is justifiably believed by the child at school. Who would
+imagine that this simple law has plunged the conscientiously thoughtful
+physicist into the greatest intellectual difficulties? Let us consider how
+these difficulties arise.
+</p>
+
+<p>
+Of course we must refer the process of the propagation of light (and indeed
+every other process) to a rigid reference-body (co-ordinate system). As such a
+system let us again choose our embankment. We shall imagine the air above it to
+have been removed. If a ray of light be sent along the embankment, we see from
+the above that the tip of the ray will be transmitted with the velocity <i>c</i>
+relative to the embankment. Now let us suppose that our railway carriage is
+again travelling along the railway lines with the velocity <i>v</i>, and that its
+direction is the same as that of the ray of light, but its velocity of course
+much less. Let us inquire about the velocity of propagation of the ray of light
+relative to the carriage. It is obvious that we can here apply the
+consideration of the previous section, since the ray of light plays the part of
+the man walking along relatively to the carriage. The velocity <i>W</i> of the man
+relative to the embankment is here replaced by the velocity of light relative
+to the embankment. <i>w</i> is the required velocity of light with respect to the
+carriage, and we have
+</p>
+
+<p>
+<i>w = c – v.</i>
+</p>
+
+<p>
+The velocity of propagation ot a ray of light relative to the carriage thus
+comes out smaller than <i>c</i>.
+</p>
+
+<p>
+But this result comes into conflict with the principle of relativity set forth
+in Section V. For, like every other general law of nature, the law of the
+transmission of light <i>in vacuo</i> [in vacuum] must, according to the principle of
+relativity, be the same for the railway carriage as reference-body as when the
+rails are the body of reference. But, from our above consideration, this would
+appear to be impossible. If every ray of light is propagated relative to the
+embankment with the velocity <i>c</i>, then for this reason it would appear that
+another law of propagation of light must necessarily hold with respect to the
+carriage—a result contradictory to the principle of relativity.
+</p>
+
+<p>
+In view of this dilemma there appears to be nothing else for it than to abandon
+either the principle of relativity or the simple law of the propagation of
+light <i>in vacuo</i>. Those of you who have carefully followed the preceding
+discussion are almost sure to expect that we should retain the principle of
+relativity, which appeals so convincingly to the intellect because it is so
+natural and simple. The law of the propagation of light <i>in vacuo</i> would then
+have to be replaced by a more complicated law conformable to the principle of
+relativity. The development of theoretical physics shows, however, that we
+cannot pursue this course. The epoch-making theoretical investigations of H. A.
+Lorentz on the electrodynamical and optical phenomena connected with moving
+bodies show that experience in this domain leads conclusively to a theory of
+electromagnetic phenomena, of which the law of the constancy of the velocity of
+light in vacuo is a necessary consequence. Prominent theoretical physicists
+were therefore more inclined to reject the principle of relativity, in spite of
+the fact that no empirical data had been found which were contradictory to this
+principle.
+</p>
+
+<p>
+At this juncture the theory of relativity entered the arena. As a result of an
+analysis of the physical conceptions of time and space, it became evident that
+<i>in reality there is not the least incompatibilitiy between the principle of
+relativity and the law of propagation of light</i>, and that by systematically
+holding fast to both these laws a logically rigid theory could be arrived at.
+This theory has been called the <i>special theory of relativity</i> to distinguish it
+from the extended theory, with which we shall deal later. In the following
+pages we shall present the fundamental ideas of the special theory of
+relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap08"></a>VIII.<br/>
+ON THE IDEA OF TIME IN PHYSICS</h3>
+
+<p>
+Lightning has struck the rails on our railway embankment at two places <i>A</i> and <i>B</i>
+far distant from each other. I make the additional assertion that these two
+lightning flashes occurred simultaneously. If I ask you whether there is sense
+in this statement, you will answer my question with a decided “Yes.”
+But if I now approach you with the request to explain to me the sense of the
+statement more precisely, you find after some consideration that the answer to
+this question is not so easy as it appears at first sight.
+</p>
+
+<p>
+After some time perhaps the following answer would occur to you: “The
+significance of the statement is clear in itself and needs no further
+explanation; of course it would require some consideration if I were to be
+commissioned to determine by observations whether in the actual case the two
+events took place simultaneously or not.” I cannot be satisfied with this
+answer for the following reason. Supposing that as a result of ingenious
+considerations an able meteorologist were to discover that the lightning must
+always strike the places <i>A</i> and <i>B</i> simultaneously, then we should be faced with
+the task of testing whether or not this theoretical result is in accordance
+with the reality. We encounter the same difficulty with all physical statements
+in which the conception “simultaneous” plays a part. The concept does
+not exist for the physicist until he has the possibility of discovering whether
+or not it is fulfilled in an actual case. We thus require a definition of
+simultaneity such that this definition supplies us with the method by means of
+which, in the present case, he can decide by experiment whether or not both the
+lightning strokes occurred simultaneously. As long as this requirement is not
+satisfied, I allow myself to be deceived as a physicist (and of course the same
+applies if I am not a physicist), when I imagine that I am able to attach a
+meaning to the statement of simultaneity. (I would ask the reader not to
+proceed farther until he is fully convinced on this point.)
+</p>
+
+<p>
+After thinking the matter over for some time you then offer the following
+suggestion with which to test simultaneity. By measuring along the rails, the
+connecting line <i>AB</i> should be measured up and an observer placed at the
+mid-point M of the distance <i>AB</i>. This observer should be supplied with an
+arrangement (<i>e.g.</i> two mirrors inclined at 90&deg;) which allows him visually to
+observe both places <i>A</i> and <i>B</i> at the same time. If the observer perceives the two
+flashes of lightning at the same time, then they are simultaneous.
+</p>
+
+<p>
+I am very pleased with this suggestion, but for all that I cannot regard the
+matter as quite settled, because I feel constrained to raise the following
+objection:
+“Your definition would certainly be right, if only I knew that the light
+by means of which the observer at <i>M</i> perceives the lightning flashes travels
+along the length <i>A</i> &#8594; <i>M</i> with the same velocity as along the length <i>B</i>
+&#8594; <i>M</i>. But an examination of this supposition would only be possible if we
+already had at our disposal the means of measuring time. It would thus appear
+as though we were moving here in a logical circle.”
+</p>
+
+<p>
+After further consideration you cast a somewhat disdainful glance at me—and
+rightly so—and you declare:
+“I maintain my previous definition nevertheless, because in reality it
+assumes absolutely nothing about light. There is only <i>one</i> demand to be made of
+the definition of simultaneity, namely, that in every real case it must supply
+us with an empirical decision as to whether or not the conception that has to
+be defined is fulfilled. That my definition satisfies this demand is
+indisputable. That light requires the same time to traverse the path <i>A</i> &#8594;
+<i>M</i> as for the path <i>B</i> &#8594; <i>M</i> is in reality neither a <i>supposition nor a
+hypothesis</i> about the physical nature of light, but a <i>stipulation</i> which I can
+make of my own freewill in order to arrive at a definition of
+simultaneity.”
+</p>
+
+<p>
+It is clear that this definition can be used to give an exact meaning not only
+to <i>two</i> events, but to as many events as we care to choose, and
+independently of the positions of the scenes of the events with respect to the
+body of reference<a href="#linknote-7" name="linknoteref-7"
+id="linknoteref-7">[7]</a> (here the railway embankment). We are thus led also
+to a definition of “time” in physics. For this purpose we suppose
+that clocks of identical construction are placed at the points <i>A, B</i> and
+<i>C</i> of the railway line (co-ordinate system) and that they are set in such
+a manner that the positions of their pointers are simultaneously (in the above
+sense) the same. Under these conditions we understand by the “time”
+of an event the reading (position of the hands) of that one of these clocks
+which is in the immediate vicinity (in space) of the event. In this manner a
+time-value is associated with every event which is essentially capable of
+observation.
+</p>
+
+<p>
+<a name="linknote-7" id="linknote-7">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-7"> [7]</a><br/> We suppose further that, when three
+events <i>A, B</i> and <i>C</i> occur in different places in such a manner
+that, if <i>A</i> is simultaneous with <i>B</i>, and <i>B</i> is simultaneous
+with <i>C</i> (simultaneous in the sense of the above definition), then the
+criterion for the simultaneity of the pair of events <i>A, C</i> is also
+satisfied. This assumption is a physical hypothesis about the law of
+propagation of light; it must certainly be fulfilled if we are to maintain the
+law of the constancy of the velocity of light <i>in vacuo</i>.
+</p>
+
+<p>
+This stipulation contains a further physical hypothesis, the validity of which
+will hardly be doubted without empirical evidence to the contrary. It has been
+assumed that all these clocks <i>go at the same rate</i> if they are of identical
+construction. Stated more exactly: When two clocks arranged at rest in
+different places of a reference-body are set in such a manner that a <i>particular</i>
+position of the pointers of the one clock is <i>simultaneous</i> (in the above sense)
+with the <i>same</i> position, of the pointers of the other clock, then identical
+“settings” are always simultaneous (in the sense of the above
+definition).
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap09"></a>IX.<br/>
+THE RELATIVITY OF SIMULTANEITY</h3>
+
+<p>
+Up to now our considerations have been referred to a particular body of
+reference, which we have styled a “railway embankment.” We suppose a
+very long train travelling along the rails with the constant velocity v and in
+the direction indicated in Fig 1. People travelling in this train will with a
+vantage view the train as a rigid reference-body (co-ordinate system); they
+regard all events in reference to the train. Then every event which takes place
+along the line also takes place at a particular point of the train. Also the
+definition of simultaneity can be given relative to the train in exactly the
+same way as with respect to the embankment. As a natural consequence, however,
+the following question arises:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image001.jpg" style="width:100%;" alt="image001" /><br/><br/>
+</div>
+
+<p>
+Are two events (<i>e.g.</i> the two strokes of lightning <i>A</i> and <i>B</i>) which are
+simultaneous <i>with reference to the railway embankment</i> also simultaneous
+<i>relatively to the train?</i> We shall show directly that the answer must be in the
+negative.
+</p>
+
+<p>
+When we say that the lightning strokes <i>A</i> and <i>B</i> are simultaneous with respect to
+be embankment, we mean: the rays of light emitted at the places <i>A</i> and <i>B</i>, where
+the lightning occurs, meet each other at the mid-point <i>M</i> of the length
+<i>A</i> &#8594; <i>B</i> of the embankment. But the events <i>A</i> and <i>B</i> also correspond
+to positions <i>A</i> and <i>B</i> on the train. Let <i>M&#x2032;</i> be the mid-point of the distance <i>A</i>
+&#8594; <i>B</i> on the travelling train. Just when the flashes (as judged from the
+embankment) of lightning occur, this point <i>M&#x2032;</i> naturally coincides with the
+point <i>M</i> but it moves towards the right in the diagram with the velocity v of
+the train. If an observer sitting in the position <i>M&#x2032;</i> in the train did not
+possess this velocity, then he would remain permanently at M, and the light
+rays emitted by the flashes of lightning <i>A</i> and <i>B</i> would reach him
+simultaneously, <i>i.e.</i> they would meet just where he is situated. Now in reality
+(considered with reference to the railway embankment) he is hastening towards
+the beam of light coming from <i>B</i>, whilst he is riding on ahead of the beam of
+light coming from <i>A</i>. Hence the observer will see the beam of light emitted from
+<i>B</i> earlier than he will see that emitted from <i>A</i>. Observers who take the railway
+train as their reference-body must therefore come to the conclusion that the
+lightning flash <i>B</i> took place earlier than the lightning flash <i>A</i>. We thus arrive
+at the important result:
+</p>
+
+<p>
+Events which are simultaneous with reference to the embankment are not
+simultaneous with respect to the train, and <i>vice versa</i> (relativity of
+simultaneity). Every reference-body (co-ordinate system) has its own particular
+time; unless we are told the reference-body to which the statement of time
+refers, there is no meaning in a statement of the time of an event.
+</p>
+
+<p>
+Now before the advent of the theory of relativity it had always tacitly been
+assumed in physics that the statement of time had an absolute significance,
+<i>i.e.</i> that it is independent of the state of motion of the body of reference.
+But we have just seen that this assumption is incompatible with the most
+natural definition of simultaneity; if we discard this assumption, then the
+conflict between the law of the propagation of light <i>in vacuo</i> and the principle
+of relativity (developed in Section VII) disappears.
+</p>
+
+<p>
+We were led to that conflict by the considerations of Section VI, which are now
+no longer tenable. In that section we concluded that the man in the carriage,
+who traverses the distance <i>w per second</i> relative to the carriage, traverses the
+same distance also with respect to the embankment <i>in each second</i> of time. But,
+according to the foregoing considerations, the time required by a particular
+occurrence with respect to the carriage must not be considered equal to the
+duration of the same occurrence as judged from the embankment (as
+reference-body). Hence it cannot be contended that the man in walking travels
+the distance <i>w</i> relative to the railway line in a time which is equal to one
+second as judged from the embankment.
+</p>
+
+<p>
+Moreover, the considerations of Section VI are based on yet a second assumption,
+which, in the light of a strict consideration, appears to be arbitrary,
+although it was always tacitly made even before the introduction of the theory
+of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap10"></a>X.<br/>
+ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE</h3>
+
+<p>
+Let us consider two particular points on the train <a href="#linknote-8" name="linknoteref-8" id="linknoteref-8">[8]</a> travelling
+along the embankment with the velocity <i>v</i>, and inquire as to their distance
+apart. We already know that it is necessary to have a body of reference for the
+measurement of a distance, with respect to which body the distance can be
+measured up. It is the simplest plan to use the train itself as reference-body
+(co-ordinate system). An observer in the train measures the interval by marking
+off his measuring-rod in a straight line (<i>e.g.</i> along the floor of the carriage)
+as many times as is necessary to take him from the one marked point to the
+other. Then the number which tells us how often the rod has to be laid down is
+the required distance.
+</p>
+
+<p>
+<a name="linknote-8" id="linknote-8">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-8"> [8]</a><br/> <i>e.g.</i> the middle of the first and
+of the hundredth carriage.
+</p>
+
+<p>
+It is a different matter when the distance has to be judged from the railway
+line. Here the following method suggests itself. If we call <i>A&#x2032;</i> and <i>B&#x2032;</i> the two
+points on the train whose distance apart is required, then both of these points
+are moving with the velocity v along the embankment. In the first place we
+require to determine the points <i>A</i> and <i>B</i> of the embankment which are just being
+passed by the two points <i>A&#x2032;</i> and <i>B&#x2032;</i> at a particular time t—judged from the
+embankment. These points <i>A</i> and <i>B</i> of the embankment can be determined by
+applying the definition of time given in Section VIII. The distance between these
+points A and B is then measured by repeated application of the measuring-rod
+along the embankment.
+</p>
+
+<p>
+<i>A priori</i> it is by no means certain that this last measurement will supply us
+with the same result as the first. Thus the length of the train as measured
+from the embankment may be different from that obtained by measuring in the
+train itself. This circumstance leads us to a second objection which must be
+raised against the apparently obvious consideration of Section VI. Namely, if
+the man in the carriage covers the distance <i>w</i> in a unit of time—<i>measured from
+the train</i>,—then this distance—<i>as measured from the embankment</i> is not
+necessarily also equal to <i>w</i>.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap11"></a>XI.<br/>
+THE LORENTZ TRANSFORMATION</h3>
+
+<p>
+The results of the last three sections show that the apparent incompatibility
+of the law of propagation of light with the principle of relativity (Section VII)
+has been derived by means of a consideration which borrowed two unjustifiable
+hypotheses from classical mechanics; these are as follows:
+</p>
+
+<p class="letter">
+(1) The time-interval (time) between two events is independent of the condition
+of motion of the body of reference.
+</p>
+
+<p class="letter">
+(2) The space-interval (distance) between two points of a rigid body is
+independent of the condition of motion of the body of reference.
+</p>
+
+<p>
+If we drop these hypotheses, then the dilemma of Section VII disappears, because
+the theorem of the addition of velocities derived in Section VI becomes invalid.
+The possibility presents itself that the law of the propagation of light <i>in
+vacuo</i> may be compatible with the principle of relativity, and the question
+arises: How have we to modify the considerations of Section VI in order to
+remove the apparent disagreement between these two fundamental results of
+experience? This question leads to a general one. In the discussion of Section
+VI we have to do with places and times relative both to the train and to the
+embankment. How are we to find the place and time of an event in relation to
+the train, when we know the place and time of the event with respect to the
+railway embankment? Is there a thinkable answer to this question of such a
+nature that the law of transmission of light <i>in vacuo</i> does not contradict the
+principle of relativity? In other words: Can we conceive of a relation between
+place and time of the individual events relative to both reference-bodies, such
+that every ray of light possesses the velocity of transmission <i>c</i> relative to
+the embankment and relative to the train? This question leads to a quite
+definite positive answer, and to a perfectly definite transformation law for
+the space-time magnitudes of an event when changing over from one body of
+reference to another.
+</p>
+
+<p>
+Before we deal with this, we shall introduce the following incidental
+consideration. Up to the present we have only considered events taking place
+along the embankment, which had mathematically to assume the function of a
+straight line. In the manner indicated in Section II we can imagine this
+reference-body supplemented laterally and in a vertical direction by means of a
+framework of rods, so that an event which takes place anywhere can be localised
+with reference to this framework.
+Similarly, we can imagine the train travelling with the velocity <i>v</i> to be
+continued across the whole of space, so that every event, no matter how far off
+it may be, could also be localised with respect to the second framework.
+Without committing any fundamental error, we can disregard the fact that in
+reality these frameworks would continually interfere with each other, owing to
+the impenetrability of solid bodies. In every such framework we imagine three
+surfaces perpendicular to each other marked out, and designated as
+“co-ordinate planes” (“co-ordinate system”). A co-ordinate
+system <i>K</i> then corresponds to the embankment, and a co-ordinate system <i>K&#x2032;</i> to the
+train. An event, wherever it may have taken place, would be fixed in space with
+respect to <i>K</i> by the three perpendiculars <i>x, y, z</i> on the co-ordinate planes, and
+with regard to time by a time value <i>t</i>. Relative to <i>K&#x2032;, the same event</i> would be
+fixed in respect of space and time by corresponding values <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>,
+which of course are not identical with <i>x, y, z, t</i>. It has already been set
+forth in detail how these magnitudes are to be regarded as results of physical
+measurements.
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image002.jpg" style="width:100%;" alt="image002" /><br/><br/>
+</div>
+
+<p>
+Obviously our problem can be exactly formulated in the following manner. What
+are the values <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>, of an event with respect to <i>K&#x2032;</i>, when the
+magnitudes <i>x, y, z, t</i>, of the same event with respect to <i>K</i> are given? The
+relations must be so chosen that the law of the transmission of light in vacuo
+is satisfied for one and the same ray of light (and of course for every ray)
+with respect to <i>K</i> and <i>K&#x2032;</i>. For the relative orientation in space of the
+co-ordinate systems indicated in the diagram (Fig. 2), this problem is solved
+by means of the equations:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image003.jpg" style="width:100%;" alt="image003" /><br/><br/>
+</div>
+
+<p class="center">
+<i>y&#x2032;</i> = <i>y</i>
+</p>
+<p class="center">
+<i>z&#x2032;</i> = <i>z</i>
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image004.jpg" style="width:100%;" alt="image004" /><br/><br/>
+</div>
+
+<p>
+This system of equations is known as the “Lorentz
+transformation.”<a href="#linknote-9" name="linknoteref-9" id="linknoteref-9">[9]</a>
+</p>
+
+<p>
+<a name="linknote-9" id="linknote-9">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-9"> [9]</a><br/> A simple derivation of the Lorentz
+transformation is given in Appendix I.
+</p>
+
+<p>
+If in place of the law of transmission of light we had taken as our basis the
+tacit assumptions of the older mechanics as to the absolute character of times
+and lengths, then instead of the above we should have obtained the following
+equations:
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>x</i> – <i>vt</i>
+</p>
+
+<p class="center">
+<i>y&#x2032;</i> = <i>y</i>
+</p>
+
+<p class="center">
+<i>z&#x2032;</i> = <i>z</i>
+</p>
+
+<p class="center">
+<i>t&#x2032;</i> = <i>t</i>
+</p>
+
+<p>
+This system of equations is often termed the “Galilei
+transformation.” The Galilei transformation can be obtained from the
+Lorentz transformation by substituting an infinitely large value for the
+velocity of light <i>c</i> in the latter transformation.
+</p>
+
+<p>
+Aided by the following illustration, we can readily see that, in accordance
+with the Lorentz transformation, the law of the transmission of light <i>in vacuo</i>
+is satisfied both for the reference-body <i>K</i> and for the reference-body <i>K&#x2032;</i>. A
+light-signal is sent along the positive <i>x</i>-axis, and this light-stimulus
+advances in accordance with the equation
+</p>
+
+<p class="center">
+<i>x</i> = <i>ct</i>,
+</p>
+
+<p class="noindent">
+<i>i.e.</i> with the velocity <i>c</i>. According to the equations of the Lorentz
+transformation, this simple relation between <i>x</i> and <i>t</i> involves a relation
+between <i>x&#x2032;</i> and <i>t&#x2032;</i>. In point of fact, if we substitute for <i>x</i> the value <i>ct</i> in the
+first and fourth equations of the Lorentz transformation, we obtain:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image005.jpg" style="width:100%;" alt="image005" /><br/><br/>
+</div>
+
+<p class="noindent">
+from which, by division, the expression
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>ct&#x2032;</i>
+</p>
+
+<p class="noindent">
+immediately follows. If referred to the system <i>K&#x2032;</i>, the propagation of light
+takes place according to this equation. We thus see that the velocity of
+transmission relative to the reference-body <i>K&#x2032;</i> is also equal to <i>c</i>. The same
+result is obtained for rays of light advancing in any other direction
+whatsoever. Of cause this is not surprising, since the equations of the Lorentz
+transformation were derived conformably to this point of view.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap12"></a>XII.<br/>
+THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION</h3>
+
+<p>
+Place a metre-rod in the <i>x&#x2032;</i>-axis of <i>K&#x2032;</i> in such a manner that one end (the
+beginning) coincides with the point <i>x&#x2032;</i> = 0 whilst the other end (the end of the
+rod) coincides with the point <i>x&#x2032;</i> = 1. What is the length of the metre-rod
+relatively to the system <i>K</i>? In order to learn this, we need only ask where the
+beginning of the rod and the end of the rod lie with respect to <i>K</i> at a
+particular time <i>t</i> of the system <i>K</i>. By means of the first equation of the
+Lorentz transformation the values of these two points at the time <i>t</i> = 0 can be
+shown to be
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image006.jpg" style="width:100%;" alt="image006" /><br/><br/>
+</div>
+
+<p class="noindent">
+the distance between the points being
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image007.jpg" style="width:100%;" alt="image007" /><br/><br/>
+</div>
+
+<p class="noindent">
+But the metre-rod is moving with the velocity <i>v</i> relative to <i>K</i>. It therefore
+follows that the length of a rigid metre-rod moving in the direction of its
+length with a velocity <i>v</i> is
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image008.jpg" style="width:100%;" alt="image008" /><br/><br/>
+</div>
+
+<p class="noindent">
+of a metre. The rigid rod is thus shorter when in motion than when at rest, and
+the more quickly it is moving, the shorter is the rod. For the velocity <i>v</i> = <i>c</i> we
+should have
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image009.jpg" style="width:100%;" alt="image009" /><br/><br/>
+</div>
+
+<p class="noindent">
+and for still greater velocities the square-root becomes imaginary. From this
+we conclude that in the theory of relativity the velocity <i>c</i> plays the
+part of a limiting velocity, which can neither be reached nor exceeded by any
+real body.
+</p>
+
+<p>
+Of course this feature of the velocity <i>c</i> as a limiting velocity also clearly
+follows from the equations of the Lorentz transformation, for these became
+meaningless if we choose values of <i>v</i> greater than <i>c</i>.
+</p>
+
+<p>
+If, on the contrary, we had considered a metre-rod at rest in the <i>x</i>-axis with
+respect to <i>K</i>, then we should have found that the length of the rod as judged
+from <i>K&#x2032;</i> would have been
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image010.jpg" style="width:100%;" alt="image010" /><br/><br/>
+</div>
+
+<p class="noindent">
+this is quite in accordance with the principle of relativity which forms the
+basis of our considerations.
+</p>
+
+<p>
+<i>A priori</i> it is quite clear that we must be able to learn something about the
+physical behaviour of measuring-rods and clocks from the equations of
+transformation, for the magnitudes <i>z, y, x, t</i>, are nothing more nor less than
+the results of measurements obtainable by means of measuring-rods and clocks.
+If we had based our considerations on the Galileian transformation we should
+not have obtained a contraction of the rod as a consequence of its motion.
+</p>
+
+<p>
+Let us now consider a seconds-clock which is permanently situated at the origin
+(<i>x&#x2032;</i> = 0) of <i>K&#x2032;</i>. <i>t&#x2032;</i> = 0 and <i>t&#x2032;</i> = 1 are two successive ticks of this clock. The first
+and fourth equations of the Lorentz transformation give for these two ticks:
+</p>
+
+<p>
+<i>t</i> = 0
+</p>
+
+<p class="noindent">
+and
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image011.jpg" style="width:100%;" alt="image011" /><br/><br/>
+</div>
+
+<p>
+As judged from <i>K</i>, the clock is moving with the velocity <i>v</i>; as judged from this
+reference-body, the time which elapses between two strokes of the clock is not
+one second, but
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image012.jpg" style="width:100%;" alt="image012" /><br/><br/>
+</div>
+
+<p class="noindent">
+seconds, <i>i.e.</i> a somewhat larger time. As a consequence of its motion the clock
+goes more slowly than when at rest. Here also the velocity <i>c</i> plays the part of
+an unattainable limiting velocity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap13"></a>XIII.<br/>
+THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU</h3>
+
+<p>
+Now in practice we can move clocks and measuring-rods only with velocities that
+are small compared with the velocity of light; hence we shall hardly be able to
+compare the results of the previous section directly with the reality. But, on
+the other hand, these results must strike you as being very singular, and for
+that reason I shall now draw another conclusion from the theory, one which can
+easily be derived from the foregoing considerations, and which has been most
+elegantly confirmed by experiment.
+</p>
+
+<p>
+In Section VI we derived the theorem of the addition of velocities in one
+direction in the form which also results from the hypotheses of classical
+mechanics. This theorem can also be deduced readily from the Galilei
+transformation (Section XI). In place of the man walking inside the carriage,
+we introduce a point moving relatively to the co-ordinate system <i>K&#x2032;</i> in
+accordance with the equation
+</p>
+
+<p>
+<i>x&#x2032;</i> = <i>wt&#x2032;</i>
+</p>
+
+<p class="noindent">
+By means of the first and fourth equations of the Galilei transformation we can
+express <i>x&#x2032;</i> and <i>t&#x2032;</i> in terms of <i>x</i> and <i>t</i>, and we then obtain
+</p>
+
+<p>
+<i>x</i> = (<i>v</i> + <i>w</i>)<i>t</i>
+</p>
+
+<p class="noindent">
+This equation expresses nothing else than the law of motion of the point with
+reference to the system <i>K</i> (of the man with reference to the embankment). We
+denote this velocity by the symbol <i>W</i>, and we then obtain, as in Section VI,
+</p>
+
+<p>
+<i>W</i> = <i>v</i> + <i>w</i> . . . . . . . (A).
+</p>
+
+<p>
+But we can carry out this consideration just as well on the basis of the theory
+of relativity. In the equation
+</p>
+
+<p>
+<i>x&#x2032;</i> = <i>wt&#x2032;</i>
+</p>
+
+<p>
+we must then express <i>x&#x2032;</i> and <i>t&#x2032;</i> in terms of <i>x</i> and <i>t</i>, making use of the first and
+fourth equations of the <i>Lorentz transformation</i>. Instead of the equation (A) we
+then obtain the equation
+</p>
+
+<div class="fig" style="width:50%;">
+<img src="images/image013.jpg" style="width:100%;" alt="image013" /><br/><br/>
+</div>
+
+<p class="noindent">
+which corresponds to the theorem of addition for velocities in one direction
+according to the theory of relativity. The question now arises as to which of
+these two theorems is the better in accord with experience. On this point we
+are enlightened by a most important experiment which the brilliant physicist
+Fizeau performed more than half a century ago, and which has been repeated
+since then by some of the best experimental physicists, so that there can be no
+doubt about its result. The experiment is concerned with the following
+question. Light travels in a motionless liquid with a particular velocity
+<i>w</i>. How quickly does it travel in the direction of the arrow in the tube
+<i>T</i> (see the accompanying diagram, Fig. 3) when the liquid above mentioned
+is flowing through the tube with a velocity <i>v</i>?
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image014.jpg" style="width:100%;" alt="image014" /><br/><br/>
+</div>
+
+<p>
+In accordance with the principle of relativity we shall certainly have to take
+for granted that the propagation of light always takes place with the same
+velocity <i>w with respect to the liquid</i>, whether the latter is in motion with
+reference to other bodies or not. The velocity of light relative to the liquid
+and the velocity of the latter relative to the tube are thus known, and we
+require the velocity of light relative to the tube.
+</p>
+
+<p>
+It is clear that we have the problem of Section VI again before us. The tube
+plays the part of the railway embankment or of the co-ordinate system <i>K</i>,
+the liquid plays the part of the carriage or of the co-ordinate system
+<i>K&#x2032;</i>, and finally, the light plays the part of the man walking
+along the carriage, or of the moving point in the present section. If we denote
+the velocity of the light relative to the tube by <i>W</i>, then this is given
+by the equation (A) or (B), according as the Galilei transformation or the
+Lorentz transformation corresponds to the facts. Experiment<a
+href="#linknote-10" name="linknoteref-10" id="linknoteref-10">[10]</a> decides
+in favour of equation (B) derived from the theory of relativity, and the
+agreement is, indeed, very exact. According to recent and most excellent
+measurements by Zeeman, the influence of the velocity of flow <i>v</i> on the
+propagation of light is represented by formula (B) to within one per cent.
+</p>
+
+<p>
+<a name="linknote-10" id="linknote-10">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-10"> [10]</a><br/> Fizeau found
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image015.jpg" style="width:100%;" alt="image015" /><br/><br/>
+</div>
+
+<p class="footnote">
+where
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image016.jpg" style="width:100%;" alt="image016" /><br/><br/>
+</div>
+
+<p class="footnote">
+is the index of refraction of the liquid. On the other hand, owing to the
+smallness of
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image017.jpg" style="width:100%;" alt="image017" /><br/><br/>
+</div>
+
+<p class="footnote">
+as compared with 1, we can replace (B) in the first place by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image018.jpg" style="width:100%;" alt="image018" /><br/><br/>
+</div>
+
+<p class="footnote">
+or to the same order of approximation by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image019.jpg" style="width:100%;" alt="image019" /><br/><br/>
+</div>
+
+<p class="footnote">
+which agrees with Fizeau’s result.
+</p>
+
+<p>
+Nevertheless we must now draw attention to the fact that a theory of this
+phenomenon was given by H. A. Lorentz long before the statement of the theory
+of relativity. This theory was of a purely electrodynamical nature, and was
+obtained by the use of particular hypotheses as to the electromagnetic
+structure of matter. This circumstance, however, does not in the least diminish
+the conclusiveness of the experiment as a crucial test in favour of the theory
+of relativity, for the electrodynamics of Maxwell-Lorentz, on which the
+original theory was based, in no way opposes the theory of relativity. Rather
+has the latter been developed trom electrodynamics as an astoundingly simple
+combination and generalisation of the hypotheses, formerly independent of each
+other, on which electrodynamics was built.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap14"></a>XIV.<br/>
+THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY</h3>
+
+<p>
+Our train of thought in the foregoing pages can be epitomised in the following
+manner. Experience has led to the conviction that, on the one hand, the
+principle of relativity holds true and that on the other hand the velocity of
+transmission of light <i>in vacuo</i> has to be considered equal to a constant <i>c</i>. By
+uniting these two postulates we obtained the law of transformation for the
+rectangular co-ordinates <i>x, y, z</i> and the time <i>t</i> of the events which constitute
+the processes of nature. In this connection we did not obtain the Galilei
+transformation, but, differing from classical mechanics, the <i>Lorentz
+transformation</i>.
+</p>
+
+<p>
+The law of transmission of light, the acceptance of which is justified by our
+actual knowledge, played an important part in this process of thought. Once in
+possession of the Lorentz transformation, however, we can combine this with the
+principle of relativity, and sum up the theory thus:
+</p>
+
+<p>
+Every general law of nature must be so constituted that it is transformed into
+a law of exactly the same form when, instead of the space-time variables <i>x, y,
+z, t</i> of the original coordinate system <i>K</i>, we introduce new space-time variables
+<i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i> of a co-ordinate system <i>K&#x2032;</i>. In this connection the relation
+between the ordinary and the accented magnitudes is given by the Lorentz
+transformation. Or in brief: General laws of nature are co-variant with
+respect to Lorentz transformations.
+</p>
+
+<p>
+This is a definite mathematical condition that the theory of relativity demands
+of a natural law, and in virtue of this, the theory becomes a valuable
+heuristic aid in the search for general laws of nature. If a general law of
+nature were to be found which did not satisfy this condition, then at least one
+of the two fundamental assumptions of the theory would have been disproved. Let
+us now examine what general results the latter theory has hitherto evinced.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap15"></a>XV.<br/>
+GENERAL RESULTS OF THE THEORY</h3>
+
+<p>
+It is clear from our previous considerations that the (special) theory of
+relativity has grown out of electrodynamics and optics. In these fields it has
+not appreciably altered the predictions of theory, but it has considerably
+simplified the theoretical structure, <i>i.e.</i> the derivation of laws, and—what is
+incomparably more important—it has considerably reduced the number of
+independent hypotheses forming the basis of theory. The special theory of
+relativity has rendered the Maxwell-Lorentz theory so plausible, that the
+latter would have been generally accepted by physicists even if experiment had
+decided less unequivocally in its favour.
+</p>
+
+<p>
+Classical mechanics required to be modified before it could come into line with
+the demands of the special theory of relativity. For the main part, however,
+this modification affects only the laws for rapid motions, in which the
+velocities of matter <i>v</i> are not very small as compared with the velocity of
+light. We have experience of such rapid motions only in the case of electrons
+and ions; for other motions the variations from the laws of classical mechanics
+are too small to make themselves evident in practice. We shall not consider the
+motion of stars until we come to speak of the general theory of relativity. In
+accordance with the theory of relativity the kinetic energy of a material point
+of mass <i>m</i> is no longer given by the well-known expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image020.jpg" style="width:100%;" alt="image020" /><br/><br/>
+</div>
+
+<p class="noindent">
+but by the expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image021.jpg" style="width:100%;" alt="image021" /><br/><br/>
+</div>
+
+<p class="noindent">
+This expression approaches infinity as the velocity <i>v</i> approaches the velocity
+of light <i>c</i>. The velocity must therefore always remain less than <i>c</i>, however
+great may be the energies used to produce the acceleration. If we develop the
+expression for the kinetic energy in the form of a series, we obtain
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image022.jpg" style="width:100%;" alt="image022" /><br/><br/>
+</div>
+
+<p>
+When
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image023.jpg" style="width:100%;" alt="image023" /><br/><br/>
+</div>
+
+<p class="noindent">
+is small compared with unity, the third of these terms is always small in
+comparison with the second, which last is alone considered in classical
+mechanics. The first term <i>mc</i><sup>2</sup> does not contain the velocity, and
+requires no consideration if we are only dealing with the question as to how
+the energy of a point-mass; depends on the velocity. We shall speak of its
+essential significance later.
+</p>
+
+<p>
+The most important result of a general character to which the special theory of
+relativity has led is concerned with the conception of mass. Before the advent
+of relativity, physics recognised two conservation laws of fundamental
+importance, namely, the law of the conservation of energy and the law of the
+conservation of mass these two fundamental laws appeared to be quite
+independent of each other. By means of the theory of relativity they have been
+united into one law. We shall now briefly consider how this unification came
+about, and what meaning is to be attached to it.
+</p>
+
+<p>
+The principle of relativity requires that the law of the conservation of energy
+should hold not only with reference to a co-ordinate system <i>K</i>, but also with
+respect to every co-ordinate system <i>K&#x2032;</i> which is in a state of uniform motion of
+translation relative to <i>K</i>, or, briefly, relative to every “Galileian”
+system of co-ordinates. In contrast to classical mechanics; the Lorentz
+transformation is the deciding factor in the transition from one such system to
+another.
+</p>
+
+<p>
+By means of comparatively simple considerations we are led to draw the
+following conclusion from these premises, in conjunction with the fundamental
+equations of the electrodynamics of Maxwell: A body moving with the velocity <i>v</i>,
+which absorbs<a href="#linknote-11" name="linknoteref-11" id="linknoteref-11">[11]</a> an amount of energy <i>E</i><sub>0</sub> in the form of
+radiation without suffering an alteration in velocity in the process, has, as a
+consequence, its energy increased by an amount
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image024.jpg" style="width:100%;" alt="image024" /><br/><br/>
+</div>
+
+<p>
+<a name="linknote-11" id="linknote-11">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-11"> [11]</a><br/> <i>E</i><sub>0</sub> is the energy
+taken up, as judged from a co-ordinate system moving with the body.
+</p>
+
+<p>
+In consideration of the expression given above for the kinetic energy of the
+body, the required energy of the body comes out to be
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image025.jpg" style="width:100%;" alt="image025" /><br/><br/>
+</div>
+
+<p>
+Thus the body has the same energy as a body of mass
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image026.jpg" style="width:100%;" alt="image026" /><br/><br/>
+</div>
+
+<p class="noindent">
+moving with the velocity <i>v</i>. Hence we can say: If a body takes up an amount of
+energy <i>E</i><sub>0</sub>, then its inertial mass increases by an amount
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image027.jpg" style="width:100%;" alt="image027" /><br/><br/>
+</div>
+
+<p class="noindent">
+the inertial mass of a body is not a constant but varies according to the
+change in the energy of the body. The inertial mass of a system of bodies can
+even be regarded as a measure of its energy. The law of the conservation of the
+mass of a system becomes identical with the law of the conservation of energy,
+and is only valid provided that the system neither takes up nor sends out
+energy. Writing the expression for the energy in the form
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image028.jpg" style="width:100%;" alt="image028" /><br/><br/>
+</div>
+
+<p class="noindent">
+we see that the term <i>mc</i><sup>2</sup>, which has hitherto attracted our
+attention, is nothing else than the energy possessed by the body<a href="#linknote-12" name="linknoteref-12" id="linknoteref-12">[12]</a>
+before it absorbed the energy <i>E</i><sub>0</sub>.
+</p>
+
+<p>
+<a name="linknote-12" id="linknote-12">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-12"> [12]</a><br/> As judged from a co-ordinate system
+moving with the body.
+</p>
+
+<p>
+A direct comparison of this relation with experiment is not possible at the
+present time (1920; see<a href="#linknote-Note" name="linknoteref-Note" id="linknoteref-Note">[Note]</a>, p. 48), owing to the fact that the
+changes in energy <i>E</i><sub>0</sub> to which we can subject a system are not large
+enough to make themselves perceptible as a change in the inertial mass of the
+system.
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image027.jpg" style="width:100%;" alt="image027" /><br/><br/>
+</div>
+
+<p class="noindent">
+is too small in comparison with the mass <i>m</i>, which was present before the
+alteration of the energy. It is owing to this circumstance that classical
+mechanics was able to establish successfully the conservation of mass as a law
+of independent validity.
+</p>
+
+<p>
+<a name="linknote-Note" id="linknote-Note">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-Note"> [Note]</a><br/> The equation E = mc<sup>2</sup>
+has been thoroughly proved time and again since this time.
+</p>
+
+<p>
+Let me add a final remark of a fundamental nature. The success of the
+Faraday-Maxwell interpretation of electromagnetic action at a distance resulted
+in physicists becoming convinced that there are no such things as instantaneous
+actions at a distance (not involving an intermediary medium) of the type of
+Newton’s law of gravitation.
+</p>
+
+<p>
+According to the theory of relativity, action at a distance with the velocity
+of light always takes the place of instantaneous action at a distance or of
+action at a distance with an infinite velocity of transmission. This is
+connected with the fact that the velocity <i>c</i> plays a fundamental role in this
+theory. In Part II we shall see in what way this result becomes modified in the
+general theory of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap16"></a>XVI.<br/>
+EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY</h3>
+
+<p>
+To what extent is the special theory of relativity supported by experience?
+This question is not easily answered for the reason already mentioned in
+connection with the fundamental experiment of Fizeau. The special theory of
+relativity has crystallised out from the Maxwell-Lorentz theory of
+electromagnetic phenomena. Thus all facts of experience which support the
+electromagnetic theory also support the theory of relativity. As being of
+particular importance, I mention here the fact that the theory of relativity
+enables us to predict the effects produced on the light reaching us from the
+fixed stars. These results are obtained in an exceedingly simple manner, and
+the effects indicated, which are due to the relative motion of the earth with
+reference to those fixed stars are found to be in accord with experience. We
+refer to the yearly movement of the apparent position of the fixed stars
+resulting from the motion of the earth round the sun (aberration), and to the
+influence of the radial components of the relative motions of the fixed stars
+with respect to the earth on the colour of the light reaching us from them. The
+latter effect manifests itself in a slight displacement of the spectral lines
+of the light transmitted to us from a fixed star, as compared with the position
+of the same spectral lines when they are produced by a terrestrial source of
+light (Doppler principle). The experimental arguments in favour of the
+Maxwell-Lorentz theory, which are at the same time arguments in favour of the
+theory of relativity, are too numerous to be set forth here. In reality they
+limit the theoretical possibilities to such an extent, that no other theory
+than that of Maxwell and Lorentz has been able to hold its own when tested by
+experience.
+</p>
+
+<p>
+But there are two classes of experimental facts hitherto obtained which can be
+represented in the Maxwell-Lorentz theory only by the introduction of an
+auxiliary hypothesis, which in itself—<i>i.e.</i> without making use of the theory of
+relativity—appears extraneous.
+</p>
+
+<p>
+It is known that cathode rays and the so-called &#946;-rays emitted by
+radioactive substances consist of negatively electrified particles (electrons)
+of very small inertia and large velocity. By examining the deflection of these
+rays under the influence of electric and magnetic fields, we can study the law
+of motion of these particles very exactly.
+</p>
+
+<p>
+In the theoretical treatment of these electrons, we are faced with the
+difficulty that electrodynamic theory of itself is unable to give an account of
+their nature. For since electrical masses of one sign repel each other, the
+negative electrical masses constituting the electron would necessarily be
+scattered under the influence of their mutual repulsions, unless there are
+forces of another kind operating between them, the nature of which has hitherto
+remained obscure to us.<a href="#linknote-13" name="linknoteref-13" id="linknoteref-13">[13]</a> If we now assume that the relative
+distances between the electrical masses constituting the electron remain
+unchanged during the motion of the electron (rigid connection in the sense of
+classical mechanics), we arrive at a law of motion of the electron which does
+not agree with experience. Guided by purely formal points of view, H. A.
+Lorentz was the first to introduce the hypothesis that the form of the electron
+experiences a contraction in the direction of motion in consequence of that
+motion. the contracted length being proportional to the expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image029.jpg" style="width:100%;" alt="image029" /><br/><br/>
+</div>
+
+<p class="noindent">
+This, hypothesis, which is not justifiable by any electrodynamical facts,
+supplies us then with that particular law of motion which has been confirmed
+with great precision in recent years.
+</p>
+
+<p>
+<a name="linknote-13" id="linknote-13">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-13"> [13]</a><br/> The general theory of relativity
+renders it likely that the electrical masses of an electron are held together
+by gravitational forces.
+</p>
+
+<p>
+The theory of relativity leads to the same law of motion, without requiring any
+special hypothesis whatsoever as to the structure and the behaviour of the
+electron. We arrived at a similar conclusion in Section XIII in connection with
+the experiment of Fizeau, the result of which is foretold by the theory of
+relativity without the necessity of drawing on hypotheses as to the physical
+nature of the liquid.
+</p>
+
+<p>
+The second class of facts to which we have alluded has reference to the
+question whether or not the motion of the earth in space can be made
+perceptible in terrestrial experiments. We have already remarked in Section V
+that all attempts of this nature led to a negative result. Before the theory of
+relativity was put forward, it was difficult to become reconciled to this
+negative result, for reasons now to be discussed. The inherited prejudices
+about time and space did not allow any doubt to arise as to the prime
+importance of the Galileian transformation for changing over from one body of
+reference to another. Now assuming that the Maxwell-Lorentz equations hold for
+a reference-body <i>K</i>, we then find that they do not hold for a reference-body <i>K&#x2032;</i>
+moving uniformly with respect to <i>K</i>, if we assume that the relations of the
+Galileian transformation exist between the co-ordinates of <i>K</i> and <i>K&#x2032;</i>. It thus
+appears that, of all Galileian co-ordinate systems, one (<i>K</i>) corresponding to a
+particular state of motion is physically unique. This result was interpreted
+physically by regarding <i>K</i> as at rest with respect to a hypothetical æther of
+space. On the other hand, all coordinate systems <i>K&#x2032;</i> moving relatively to <i>K</i> were
+to be regarded as in motion with respect to the æther. To this motion of <i>K&#x2032;</i>
+against the æther (“æther-drift” relative to <i>K&#x2032;</i>) were attributed the
+more complicated laws which were supposed to hold relative to <i>K&#x2032;</i>. Strictly
+speaking, such an æther-drift ought also to be assumed relative to the earth,
+and for a long time the efforts of physicists were devoted to attempts to
+detect the existence of an æther-drift at the earth’s surface.
+</p>
+
+<p>
+In one of the most notable of these attempts Michelson devised a method which
+appears as though it must be decisive. Imagine two mirrors so arranged on a
+rigid body that the reflecting surfaces face each other. A ray of light
+requires a perfectly definite time <i>T</i> to pass from one mirror to the other and
+back again, if the whole system be at rest with respect to the æther. It is
+found by calculation, however, that a slightly different time <i>T&#x2032;</i> is required
+for this process, if the body, together with the mirrors, be moving relatively
+to the æther. And yet another point: it is shown by calculation that for a
+given velocity <i>v</i> with reference to the æther, this time <i>T&#x2032;</i> is different when
+the body is moving perpendicularly to the planes of the mirrors from that
+resulting when the motion is parallel to these planes. Although the estimated
+difference between these two times is exceedingly small, Michelson and Morley
+performed an experiment involving interference in which this difference should
+have been clearly detectable. But the experiment gave a negative result—a fact
+very perplexing to physicists. Lorentz and FitzGerald rescued the theory from
+this difficulty by assuming that the motion of the body relative to the æther
+produces a contraction of the body in the direction of motion, the amount of
+contraction being just sufficient to compensate for the difference in time
+mentioned above. Comparison with the discussion in Section XII shows that also
+from the standpoint of the theory of relativity this solution of the difficulty
+was the right one. But on the basis of the theory of relativity the method of
+interpretation is incomparably more satisfactory. According to this theory
+there is no such thing as a “specially favoured” (unique) co-ordinate
+system to occasion the introduction of the æther-idea, and hence there can be
+no æther-drift, nor any experiment with which to demonstrate it. Here the
+contraction of moving bodies follows from the two fundamental principles of the
+theory, without the introduction of particular hypotheses; and as the prime
+factor involved in this contraction we find, not the motion in itself, to which
+we cannot attach any meaning, but the motion with respect to the body of
+reference chosen in the particular case in point. Thus for a co-ordinate system
+moving with the earth the mirror system of Michelson and Morley is not
+shortened, but it <i>is</i> shortened for a co-ordinate system which is at rest
+relatively to the sun.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap17"></a>XVII.<br/>
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE</h3>
+
+<p>
+The non-mathematician is seized by a mysterious shuddering when he hears of
+“four-dimensional” things, by a feeling not unlike that awakened by
+thoughts of the occult. And yet there is no more common-place statement than
+that the world in which we live is a four-dimensional space-time continuum.
+</p>
+
+<p>
+Space is a three-dimensional continuum. By this we mean that it is possible to
+describe the position of a point (at rest) by means of three numbers
+(co-ordinates) <i>x, y, z</i>, and that there is an indefinite number of points in the
+neighbourhood of this one, the position of which can be described by
+co-ordinates such as <i>x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub></i>, which may be
+as near as we choose to the respective values of the co-ordinates <i>x, y, z</i>, of
+the first point. In virtue of the latter property we speak of a
+“continuum,” and owing to the fact that there are three co-ordinates
+we speak of it as being “three-dimensional.”
+</p>
+
+<p>
+Similarly, the world of physical phenomena which was briefly called
+“world” by Minkowski is naturally four dimensional in the space-time
+sense. For it is composed of individual events, each of which is described by
+four numbers, namely, three space co-ordinates <i>x, y, z</i>, and a time co-ordinate,
+the time value <i>t</i>. The “world” is in this sense
+also a continuum; for to every event there are as many “neighbouring”
+events (realised or at least thinkable) as we care to choose, the co-ordinates
+<i>x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>, t<sub>1</sub></i> of which differ by
+an indefinitely small amount from those of the event <i>x, y, z, t</i> originally
+considered. That we have not been accustomed to regard the world in this sense
+as a four-dimensional continuum is due to the fact that in physics, before the
+advent of the theory of relativity, time played a different and more
+independent rôle, as compared with the space coordinates. It is for this reason
+that we have been in the habit of treating time as an independent continuum. As
+a matter of fact, according to classical mechanics, time is absolute, <i>i.e.</i> it
+is independent of the position and the condition of motion of the system of
+co-ordinates. We see this expressed in the last equation of the Galileian
+transformation (<i>t&#x2032;</i> = <i>t</i>).
+</p>
+
+<p>
+The four-dimensional mode of consideration of the “world” is natural
+on the theory of relativity, since according to this theory time is robbed of
+its independence. This is shown by the fourth equation of the Lorentz
+transformation:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image030.jpg" style="width:100%;" alt="image030" /><br/><br/>
+</div>
+
+<p class="noindent">
+Moreover, according to this equation the time difference &#916;<i>t&#x2032;</i> of two events
+with respect to <i>K&#x2032;</i> does not in general vanish, even when the time difference
+&#916;<i>t</i> of the same events with reference to <i>K</i> vanishes. Pure
+“space-distance” of two events with respect to <i>K</i> results in
+“time-distance ” of the same events with respect to <i>K</i>. But the
+discovery of Minkowski, which was of importance for the formal development of
+the theory of relativity, does not lie here. It is to be found rather in the
+fact of his recognition that the four-dimensional space-time continuum of the
+theory of relativity, in its most essential formal properties, shows a
+pronounced relationship to the three-dimensional continuum of Euclidean
+geometrical space.<a href="#linknote-14" name="linknoteref-14" id="linknoteref-14">[14]</a> In order to give due prominence to this
+relationship, however, we must replace the usual time co-ordinate t by an
+imaginary magnitude
+</p>
+
+<div class="fig" style="width:10%;">
+<img src="images/image031.jpg" style="width:100%;" alt="image031" /><br/><br/>
+</div>
+
+<p class="noindent">
+proportional to it. Under these conditions, the natural laws satisfying the
+demands of the (special) theory of relativity assume mathematical forms, in
+which the time co-ordinate plays exactly the same role as the three space
+co-ordinates. Formally, these four co-ordinates correspond exactly to the
+three space co-ordinates in Euclidean geometry. It must be clear even to the
+non-mathematician that, as a consequence of this purely formal addition to our
+knowledge, the theory perforce gained clearness in no mean measure.
+</p>
+
+<p>
+<a name="linknote-14" id="linknote-14">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-14"> [14]</a><br/> Cf. the somewhat more detailed
+discussion in Appendix II.
+</p>
+
+<p>
+These inadequate remarks can give the reader only a vague notion of the
+important idea contributed by Minkowski. Without it the general theory of
+relativity, of which the fundamental ideas are developed in the following
+pages, would perhaps have got no farther than its long clothes. Minkowski’s
+work is doubtless difficult of access to anyone inexperienced in mathematics,
+but since it is not necessary to have a very exact grasp of this work in order
+to understand the fundamental ideas of either the special or the general theory
+of relativity, I shall leave it here at present, and revert to it only towards
+the end of Part II.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part02"></a>PART II: THE GENERAL THEORY OF RELATIVITY</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap18"></a>XVIII.<br/>
+SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+The basal principle, which was the pivot of all our previous considerations,
+was the <i>special</i> principle of relativity, <i>i.e.</i> the principle of the physical
+relativity of all <i>uniform</i> motion. Let as once more analyse its meaning
+carefully.
+</p>
+
+<p>
+It was at all times clear that, from the point of view of the idea it conveys
+to us, every motion must be considered only as a relative motion. Returning to
+the illustration we have frequently used of the embankment and the railway
+carriage, we can express the fact of the motion here taking place in the
+following two forms, both of which are equally justifiable:
+</p>
+
+<p class="letter">
+(<i>a</i>) The carriage is in motion relative to the embankment,
+</p>
+
+<p class="letter">
+(<i>b</i>) The embankment is in motion relative to the carriage.
+</p>
+
+<p>
+In (<i>a</i>) the embankment, in (<i>b</i>) the carriage, serves as the body of reference in
+our statement of the motion taking place. If it is simply a question of
+detecting or of describing the motion involved, it is in principle immaterial
+to what reference-body we refer the motion. As already mentioned, this is
+self-evident, but it must not be confused with the much more comprehensive
+statement called “the principle of relativity,” which we have taken
+as the basis of our investigations.
+</p>
+
+<p>
+The principle we have made use of not only maintains that we may equally well
+choose the carriage or the embankment as our reference-body for the description
+of any event (for this, too, is self-evident). Our principle rather asserts
+what follows: If we formulate the general laws of nature as they are obtained
+from experience, by making use of
+</p>
+
+<p class="letter">
+(<i>a</i>) the embankment as reference-body,
+</p>
+
+<p class="letter">
+(<i>b</i>) the railway carriage as reference-body,
+</p>
+
+<p>
+then these general laws of nature (<i>e.g.</i> the laws of mechanics or the law of the
+propagation of light <i>in vacuo</i>) have exactly the same form in both cases. This
+can also be expressed as follows: For the physical description of natural
+processes, neither of the reference bodies <i>K, K&#x2032;</i> is unique (lit.
+“specially marked out”) as compared with the other. Unlike the first,
+this latter statement need not of necessity hold <i>a priori;</i> it is not contained
+in the conceptions of “motion” and “reference-body” and
+derivable from them; only <i>experience</i> can decide as to its correctness or
+incorrectness.
+</p>
+
+<p>
+Up to the present, however, we have by no means maintained the equivalence of
+<i>all</i> bodies of reference <i>K</i> in connection with the formulation of natural laws.
+Our course was more on the following Iines. In the first place, we started out
+from the assumption that there exists a reference-body <i>K</i>, whose condition of
+motion is such that the Galileian law holds with respect to it: A particle
+left to itself and sufficiently far removed from all other particles moves
+uniformly in a straight line. With reference to K (Galileian reference-body)
+the laws of nature were to be as simple as possible. But in addition to K, all
+bodies of reference <i>K&#x2032;</i> should be given preference in this sense, and they
+should be exactly equivalent to <i>K</i> for the formulation of natural laws, provided
+that they are in a state of <i>uniform rectilinear and non-rotary motion</i> with
+respect to <i>K</i>; all these bodies of reference are to be regarded as Galileian
+reference-bodies. The validity of the principle of relativity was assumed only
+for these reference-bodies, but not for others (<i>e.g.</i> those possessing motion of
+a different kind). In this sense we speak of the <i>special</i> principle of
+relativity, or special theory of relativity.
+</p>
+
+<p>
+In contrast to this we wish to understand by the “general principle of
+relativity” the following statement: All bodies of reference <i>K, K&#x2032;</i>, etc.,
+are equivalent for the description of natural phenomena (formulation of the
+general laws of nature), whatever may be their state of motion. But before
+proceeding farther, it ought to be pointed out that this formulation must be
+replaced later by a more abstract one, for reasons which will become evident at
+a later stage.
+</p>
+
+<p>
+Since the introduction of the special principle of relativity has been
+justified, every intellect which strives after generalisation must feel the
+temptation to venture the step towards the general principle of relativity. But
+a simple and apparently quite reliable consideration seems to suggest that, for
+the present at any rate, there is little hope of success in such an attempt;
+Let us imagine ourselves transferred to our old friend the railway carriage,
+which is travelling at a uniform rate. As long as it is moving uniformly, the
+occupant of the carriage is not sensible of its motion, and it is for this
+reason that he can without reluctance interpret the facts of the case as
+indicating that the carriage is at rest, but the embankment in motion.
+Moreover, according to the special principle of relativity, this interpretation
+is quite justified also from a physical point of view. If the motion of the
+carriage is now changed into a non-uniform motion, as for instance by a
+powerful application of the brakes, then the occupant of the carriage
+experiences a correspondingly powerful jerk forwards. The retarded motion is
+manifested in the mechanical behaviour of bodies relative to the person in the
+railway carriage. The mechanical behaviour is different from that of the case
+previously considered, and for this reason it would appear to be impossible
+that the same mechanical laws hold relatively to the non-uniformly moving
+carriage, as hold with reference to the carriage when at rest or in uniform
+motion. At all events it is clear that the Galileian law does not hold with
+respect to the non-uniformly moving carriage. Because of this, we feel
+compelled at the present juncture to grant a kind of absolute physical reality
+to non-uniform motion, in opposition to the general principle of relativity.
+But in what follows we shall soon see that this conclusion cannot be
+maintained.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap19"></a>XIX.<br/>
+THE GRAVITATIONAL FIELD</h3>
+
+<p>
+“If we pick up a stone and then let it go, why does it fall to the
+ground?” The usual answer to this question is: “Because it is
+attracted by the earth.” Modern physics formulates the answer rather
+differently for the following reason. As a result of the more careful study of
+electromagnetic phenomena, we have come to regard action at a distance as a
+process impossible without the intervention of some intermediary medium. If,
+for instance, a magnet attracts a piece of iron, we cannot be content to regard
+this as meaning that the magnet acts directly on the iron through the
+intermediate empty space, but we are constrained to imagine—after the
+manner of Faraday—that the magnet always calls into being something
+physically real in the space around it, that something being what we call a
+“magnetic field.” In its turn this magnetic field operates on the
+piece of iron, so that the latter strives to move towards the magnet. We shall
+not discuss here the justification for this incidental conception, which is
+indeed a somewhat arbitrary one. We shall only mention that with its aid
+electromagnetic phenomena can be theoretically represented much more
+satisfactorily than without it, and this applies particularly to the
+transmission of electromagnetic waves. The effects of gravitation also are
+regarded in an analogous manner.
+</p>
+
+<p>
+The action of the earth on the stone takes place indirectly. The earth produces
+in its surrounding a gravitational field, which acts on the stone and produces
+its motion of fall. As we know from experience, the intensity of the action on
+a body dimishes according to a quite definite law, as we proceed farther and
+farther away from the earth. From our point of view this means: The law
+governing the properties of the gravitational field in space must be a
+perfectly definite one, in order correctly to represent the diminution of
+gravitational action with the distance from operative bodies. It is something
+like this: The body (<i>e.g.</i> the earth) produces a field in its immediate
+neighbourhood directly; the intensity and direction of the field at points
+farther removed from the body are thence determined by the law which governs
+the properties in space of the gravitational fields themselves.
+</p>
+
+<p>
+In contrast to electric and magnetic fields, the gravitational field exhibits a
+most remarkable property, which is of fundamental importance for what follows.
+Bodies which are moving under the sole influence of a gravitational field
+receive an acceleration, <i>which does not in the least depend either on the
+material or on the physical state of the body.</i> For instance, a piece of lead
+and a piece of wood fall in exactly the same manner in a gravitational field
+(<i>in vacuo</i>), when they start off from rest or with the same initial velocity.
+This law, which holds most accurately, can be expressed in a different form in
+the light of the following consideration.
+</p>
+
+<p>
+According to Newton’s law of motion, we have
+</p>
+
+<p>
+(Force) = (inertial mass) x (acceleration),
+</p>
+
+<p class="noindent">
+where the “inertial mass” is a characteristic constant of the
+accelerated body. If now gravitation is the cause of the acceleration, we then
+have
+</p>
+
+<p>
+(Force) = (gravitational mass) x (intensity of the gravitational field),
+</p>
+
+<p class="noindent">
+where the “gravitational mass” is likewise a characteristic constant
+for the body. From these two relations follows:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image032.jpg" style="width:100%;" alt="image032" /><br/><br/>
+</div>
+
+<p>
+If now, as we find from experience, the acceleration is to be independent of
+the nature and the condition of the body and always the same for a given
+gravitational field, then the ratio of the gravitational to the inertial mass
+must likewise be the same for all bodies. By a suitable choice of units we can
+thus make this ratio equal to unity. We then have the following law: The
+<i>gravitational</i> mass of a body is equal to its <i>inertial</i> mass.
+</p>
+
+<p>
+It is true that this important law had hitherto been recorded in mechanics, but
+it had not been <i>interpreted</i>. A satisfactory interpretation can be obtained only
+if we recognise the following fact: <i>The same</i> quality of a body manifests
+itself according to circumstances as “inertia” or as
+“weight” (lit. “heaviness”). In the following section we
+shall show to what extent this is actually the case, and how this question is
+connected with the general postulate of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap20"></a>XX.<br/>
+THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL
+POSTULATE OF RELATIVITY</h3>
+
+<p>
+We imagine a large portion of empty space, so far removed from stars and other
+appreciable masses, that we have before us approximately the conditions
+required by the fundamental law of Galilei. It is then possible to choose a
+Galileian reference-body for this part of space (world), relative to which
+points at rest remain at rest and points in motion continue permanently in
+uniform rectilinear motion. As reference-body let us imagine a spacious chest
+resembling a room with an observer inside who is equipped with apparatus.
+Gravitation naturally does not exist for this observer. He must fasten himself
+with strings to the floor, otherwise the slightest impact against the floor
+will cause him to rise slowly towards the ceiling of the room.
+</p>
+
+<p>
+To the middle of the lid of the chest is fixed externally a hook with rope
+attached, and now a “being” (what kind of a being is immaterial to
+us) begins pulling at this with a constant force. The chest together with the
+observer then begin to move “upwards” with a uniformly accelerated
+motion. In course of time their velocity will reach unheard-of values—provided
+that we are viewing all this from another reference-body which is not being
+pulled with a rope.
+</p>
+
+<p>
+But how does the man in the chest regard the Process? The acceleration of the
+chest will be transmitted to him by the reaction of the floor of the chest. He
+must therefore take up this pressure by means of his legs if he does not wish
+to be laid out full length on the floor. He is then standing in the chest in
+exactly the same way as anyone stands in a room of a home on our earth. If he
+releases a body which he previously had in his land, the accelertion of the
+chest will no longer be transmitted to this body, and for this reason the body
+will approach the floor of the chest with an accelerated relative motion. The
+observer will further convince himself <i>that the acceleration of the body
+towards the floor of the chest is always of the same magnitude, whatever kind
+of body he may happen to use for the experiment.</i>
+</p>
+
+<p>
+Relying on his knowledge of the gravitational field (as it was discussed in the
+preceding section), the man in the chest will thus come to the conclusion that
+he and the chest are in a gravitational field which is constant with regard to
+time. Of course he will be puzzled for a moment as to why the chest does not
+fall in this gravitational field. just then, however, he discovers the hook in
+the middle of the lid of the chest and the rope which is attached to it, and he
+consequently comes to the conclusion that the chest is suspended at rest in the
+gravitational field.
+</p>
+
+<p>
+Ought we to smile at the man and say that he errs in his conclusion? I do not
+believe we ought to if we wish to remain consistent; we must rather admit that
+his mode of grasping the situation violates neither reason nor known mechanical
+laws. Even though it is being accelerated with respect to the “Galileian
+space” first considered, we can nevertheless regard the chest as being at
+rest. We have thus good grounds for extending the principle of relativity to
+include bodies of reference which are accelerated with respect to each other,
+and as a result we have gained a powerful argument for a generalised postulate
+of relativity.
+</p>
+
+<p>
+We must note carefully that the possibility of this mode of interpretation
+rests on the fundamental property of the gravitational field of giving all
+bodies the same acceleration, or, what comes to the same thing, on the law of
+the equality of inertial and gravitational mass. If this natural law did not
+exist, the man in the accelerated chest would not be able to interpret the
+behaviour of the bodies around him on the supposition of a gravitational field,
+and he would not be justified on the grounds of experience in supposing his
+reference-body to be “at rest.”
+</p>
+
+<p>
+Suppose that the man in the chest fixes a rope to the inner side of the lid,
+and that he attaches a body to the free end of the rope. The result of this
+will be to stretch the rope so that it will hang “vertically”
+downwards. If we ask for an opinion of the cause of tension in the rope, the
+man in the chest will say: “The suspended body experiences a downward
+force in the gravitational field, and this is neutralised by the tension of the
+rope; what determines the magnitude of the tension of the rope is the
+<i>gravitational mass</i> of the suspended body.” On the other hand, an observer
+who is poised freely in space will interpret the condition of things thus:
+“The rope must perforce take part in the accelerated motion of the chest,
+and it transmits this motion to the body attached to it. The tension of the
+rope is just large enough to effect the acceleration of the body. That which
+determines the magnitude of the tension of the rope is the <i>inertial mass</i> of the
+body.” Guided by this example, we see that our extension of the principle
+of relativity implies the <i>necessity</i> of the law of the equality of inertial and
+gravitational mass. Thus we have obtained a physical interpretation of this
+law.
+</p>
+
+<p>
+From our consideration of the accelerated chest we see that a general theory of
+relativity must yield important results on the laws of gravitation. In point of
+fact, the systematic pursuit of the general idea of relativity has supplied the
+laws satisfied by the gravitational field. Before proceeding farther, however,
+I must warn the reader against a misconception suggested by these
+considerations. A gravitational field exists for the man in the chest, despite
+the fact that there was no such field for the co-ordinate system first chosen.
+Now we might easily suppose that the existence of a gravitational field is
+always only an <i>apparent</i> one. We might also think that, regardless of the kind
+of gravitational field which may be present, we could always choose another
+reference-body such that <i>no</i> gravitational field exists with reference to it.
+This is by no means true for all gravitational fields, but only for those of
+quite special form. It is, for instance, impossible to choose a body of
+reference such that, as judged from it, the gravitational field of the earth
+(in its entirety) vanishes.
+</p>
+
+<p>
+We can now appreciate why that argument is not convincing, which we brought
+forward against the general principle of relativity at the end of Section XVIII.
+It is certainly true that the observer in the railway carriage experiences a
+jerk forwards as a result of the application of the brake, and that he
+recognises, in this the non-uniformity of motion (retardation) of the carriage.
+But he is compelled by nobody to refer this jerk to a “real”
+acceleration (retardation) of the carriage. He might also interpret his
+experience thus: “My body of reference (the carriage) remains permanently
+at rest. With reference to it, however, there exists (during the period of
+application of the brakes) a gravitational field which is directed forwards and
+which is variable with respect to time. Under the influence of this field, the
+embankment together with the earth moves non-uniformly in such a manner that
+their original velocity in the backwards direction is continuously
+reduced.”
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap21"></a>XXI.<br/>
+IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE SPECIAL
+THEORY OF RELATIVITY UNSATISFACTORY?</h3>
+
+<p>
+We have already stated several times that classical mechanics starts out from
+the following law: Material particles sufficiently far removed from other
+material particles continue to move uniformly in a straight line or continue in
+a state of rest. We have also repeatedly emphasised that this fundamental law
+can only be valid for bodies of reference <i>K</i> which possess certain unique states
+of motion, and which are in uniform translational motion relative to each
+other. Relative to other reference-bodies <i>K</i> the law is not valid. Both in
+classical mechanics and in the special theory of relativity we therefore
+differentiate between reference-bodies <i>K</i> relative to which the recognised
+“laws of nature” can be said to hold, and reference-bodies <i>K</i> relative
+to which these laws do not hold.
+</p>
+
+<p>
+But no person whose mode of thought is logical can rest satisfied with this
+condition of things. He asks: “How does it come that certain
+reference-bodies (or their states of motion) are given priority over other
+reference-bodies (or their states of motion)? <i>What is the reason for this
+preference?</i>” In order to show clearly what I mean by this question, I
+shall make use of a comparison.
+</p>
+
+<p>
+I am standing in front of a gas range. Standing alongside of each other on the
+range are two pans so much alike that one may be mistaken for the other. Both
+are half full of water. I notice that steam is being emitted continuously from
+the one pan, but not from the other. I am surprised at this, even if I have
+never seen either a gas range or a pan before. But if I now notice a luminous
+something of bluish colour under the first pan but not under the other, I cease
+to be astonished, even if I have never before seen a gas flame. For I can only
+say that this bluish something will cause the emission of the steam, or at
+least <i>possibly</i> it may do so. If, however, I notice the bluish something in
+neither case, and if I observe that the one continuously emits steam whilst the
+other does not, then I shall remain astonished and dissatisfied until I have
+discovered some circumstance to which I can attribute the different behaviour
+of the two pans.
+</p>
+
+<p>
+Analogously, I seek in vain for a real something in classical mechanics (or in
+the special theory of relativity) to which I can attribute the different
+behaviour of bodies considered with respect to the reference systems <i>K</i> and <i>K&#x2032;</i>.<a href="#linknote-15" name="linknoteref-15" id="linknoteref-15">[15]</a> Newton saw this objection and attempted to invalidate it, but
+without success. But E. Mach recognised it most clearly of all, and because of
+this objection he claimed that mechanics must be placed on a new basis. It can
+only be got rid of by means of a physics which is conformable to the general
+principle of relativity, since the equations of such a theory hold for every
+body of reference, whatever may be its state of motion.
+</p>
+
+<p>
+<a name="linknote-15" id="linknote-15">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-15"> [15]</a><br/> The objection is of importance more
+especially when the state of motion of the reference-body is of such a nature
+that it does not require any external agency for its maintenance, <i>e.g.</i> in the
+case when the reference-body is rotating uniformly.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap22"></a>XXII.<br/>
+A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+The considerations of Section XX show that the general principle of relativity
+puts us in a position to derive properties of the gravitational field in a
+purely theoretical manner. Let us suppose, for instance, that we know the
+space-time “course” for any natural process whatsoever, as regards
+the manner in which it takes place in the Galileian domain relative to a
+Galileian body of reference <i>K</i>. By means of purely theoretical operations
+(<i>i.e.</i> simply by calculation) we are then able to find how this known
+natural process appears, as seen from a reference-body <i>K&#x2032;</i> which is
+accelerated relatively to <i>K</i>. But since a gravitational field exists with
+respect to this new body of reference <i>K&#x2032;</i>, our consideration also teaches
+us how the gravitational field influences the process studied.
+</p>
+
+<p>
+For example, we learn that a body which is in a state of uniform rectilinear
+motion with respect to <i>K</i> (in accordance with the law of Galilei) is executing
+an accelerated and in general curvilinear motion with respect to the
+accelerated reference-body <i>K&#x2032;</i> (chest). This acceleration or curvature
+corresponds to the influence on the moving body of the gravitational field
+prevailing relatively to <i>K</i>. It is known that a gravitational field influences
+the movement of bodies in this way, so that our consideration supplies us with
+nothing essentially new.
+</p>
+
+<p>
+However, we obtain a new result of fundamental importance when we carry out the
+analogous consideration for a ray of light. With respect to the Galileian
+reference-body <i>K</i>, such a ray of light is transmitted rectilinearly with the
+velocity <i>c</i>. It can easily be shown that the path of the same ray of light is no
+longer a straight line when we consider it with reference to the accelerated
+chest (reference-body <i>K&#x2032;</i>). From this we conclude, <i>that, in general, rays of
+light are propagated curvilinearly in gravitational fields.</i> In two respects
+this result is of great importance.
+</p>
+
+<p>
+In the first place, it can be compared with the reality. Although a detailed
+examination of the question shows that the curvature of light rays required by
+the general theory of relativity is only exceedingly small for the
+gravitational fields at our disposal in practice, its estimated magnitude for
+light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of
+arc. This ought to manifest itself in the following way. As seen from the
+earth, certain fixed stars appear to be in the neighbourhood of the sun, and
+are thus capable of observation during a total eclipse of the sun. At such
+times, these stars ought to appear to be displaced outwards from the sun by an
+amount indicated above, as compared with their apparent position in the sky
+when the sun is situated at another part of the heavens. The examination of the
+correctness or otherwise of this deduction is a problem of the greatest
+importance, the early solution of which is to be expected of astronomers.<a href="#linknote-16" name="linknoteref-16" id="linknoteref-16">[16]</a>
+</p>
+
+<p>
+<a name="linknote-16" id="linknote-16">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-16"> [16]</a><br/> By means of the star photographs of
+two expeditions equipped by a Joint Committee of the Royal and Royal
+Astronomical Societies, the existence of the deflection of light demanded by
+theory was first confirmed during the solar eclipse of 29th May, 1919. (Cf.
+Appendix III.)
+</p>
+
+<p>
+In the second place our result shows that, according to the general theory of
+relativity, the law of the constancy of the velocity of light in vacuo, which
+constitutes one of the two fundamental assumptions in the special theory of
+relativity and to which we have already frequently referred, cannot claim any
+unlimited validity. A curvature of rays of light can only take place when the
+velocity of propagation of light varies with position. Now we might think that
+as a consequence of this, the special theory of relativity and with it the
+whole theory of relativity would be laid in the dust. But in reality this is
+not the case. We can only conclude that the special theory of relativity cannot
+claim an unlimited domain of validity; its results hold only so long as we are
+able to disregard the influences of gravitational fields on the phenomena (<i>e.g.</i>
+of light).
+</p>
+
+<p>
+Since it has often been contended by opponents of the theory of relativity that
+the special theory of relativity is overthrown by the general theory of
+relativity, it is perhaps advisable to make the facts of the case clearer by
+means of an appropriate comparison. Before the development of electrodynamics
+the laws of electrostatics were looked upon as the laws of electricity. At the
+present time we know that electric fields can be derived correctly from
+electrostatic considerations only for the case, which is never strictly
+realised, in which the electrical masses are quite at rest relatively to each
+other, and to the co-ordinate system. Should we be justified in saying that for
+this reason electrostatics is overthrown by the field-equations of Maxwell in
+electrodynamics? Not in the least. Electrostatics is contained in
+electrodynamics as a limiting case; the laws of the latter lead directly to
+those of the former for the case in which the fields are invariable with regard
+to time. No fairer destiny could be allotted to any physical theory, than that
+it should of itself point out the way to the introduction of a more
+comprehensive theory, in which it lives on as a limiting case.
+</p>
+
+<p>
+In the example of the transmission of light just dealt with, we have seen that
+the general theory of relativity enables us to derive theoretically the
+influence of a gravitational field on the course of natural processes, the laws
+of which are already known when a gravitational field is absent. But the most
+attractive problem, to the solution of which the general theory of relativity
+supplies the key, concerns the investigation of the laws satisfied by the
+gravitational field itself. Let us consider this for a moment.
+</p>
+
+<p>
+We are acquainted with space-time domains which behave (approximately) in a
+“Galileian” fashion under suitable choice of reference-body, <i>i.e.</i>
+domains in which gravitational fields are absent. If we now refer such a domain
+to a reference-body <i>K&#x2032;</i> possessing any kind of motion, then relative to <i>K&#x2032;</i> there
+exists a gravitational field which is variable with respect to space and time.<a href="#linknote-17" name="linknoteref-17" id="linknoteref-17">[17]</a> The character of this field will of course depend on the motion
+chosen for <i>K&#x2032;.</i> According to the general theory of relativity, the general law
+of the gravitational field must be satisfied for all gravitational fields
+obtainable in this way. Even though by no means all gravitationial fields can
+be produced in this way, yet we may entertain the hope that the general law of
+gravitation will be derivable from such gravitational fields of a special kind.
+This hope has been realised in the most beautiful manner. But between the clear
+vision of this goal and its actual realisation it was necessary to surmount a
+serious difficulty, and as this lies deep at the root of things, I dare not
+withhold it from the reader. We require to extend our ideas of the space-time
+continuum still farther.
+</p>
+
+<p>
+<a name="linknote-17" id="linknote-17">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-17"> [17]</a><br/> This follows from a generalisation of
+the discussion in Section XX.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap23"></a>XXIII.<br/>
+BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE</h3>
+
+<p>
+Hitherto I have purposely refrained from speaking about the physical
+interpretation of space- and time-data in the case of the general theory of
+relativity. As a consequence, I am guilty of a certain slovenliness of
+treatment, which, as we know from the special theory of relativity, is far from
+being unimportant and pardonable. It is now high time that we remedy this
+defect; but I would mention at the outset, that this matter lays no small
+claims on the patience and on the power of abstraction of the reader.
+</p>
+
+<p>
+We start off again from quite special cases, which we have frequently used
+before. Let us consider a space time domain in which no gravitational field
+exists relative to a reference-body <i>K</i> whose state of motion has been
+suitably chosen. <i>K</i> is then a Galileian reference-body as regards the
+domain considered, and the results of the special theory of relativity hold
+relative to <i>K</i>. Let us suppose the same domain referred to a second body
+of reference <i>K&#x2032;</i>, which is rotating uniformly with respect to <i>K</i>.
+In order to fix our ideas, we shall imagine <i>K&#x2032;</i> to be in the form of a
+plane circular disc, which rotates uniformly in its own plane about its centre.
+An observer who is sitting eccentrically on the disc <i>K&#x2032;</i> is sensible of a
+force which acts outwards in a radial direction, and which would be interpreted
+as an effect of inertia (centrifugal force) by an observer who was at rest with
+respect to the original reference-body <i>K</i>. But the observer on the disc
+may regard his disc as a reference-body which is “at rest”; on the
+basis of the general principle of relativity he is justified in doing this. The
+force acting on himself, and in fact on all other bodies which are at rest
+relative to the disc, he regards as the effect of a gravitational field.
+Nevertheless, the space-distribution of this gravitational field is of a kind
+that would not be possible on Newton’s theory of gravitation.<a
+href="#linknote-18" name="linknoteref-18" id="linknoteref-18">[18]</a> But
+since the observer believes in the general theory of relativity, this does not
+disturb him; he is quite in the right when he believes that a general law of
+gravitation can be formulated—a law which not only explains the motion of
+the stars correctly, but also the field of force experienced by himself.
+</p>
+
+<p>
+<a name="linknote-18" id="linknote-18">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-18"> [18]</a><br/> The field disappears at the centre of
+the disc and increases proportionally to the distance from the centre as we
+proceed outwards.
+</p>
+
+<p>
+The observer performs experiments on his circular disc with clocks and
+measuring-rods. In doing so, it is his intention to arrive at exact definitions
+for the signification of time- and space-data with reference to the circular
+disc <i>K&#x2032;</i>, these definitions being based on his observations. What will be his
+experience in this enterprise?
+</p>
+
+<p>
+To start with, he places one of two identically constructed clocks at the
+centre of the circular disc, and the other on the edge of the disc, so that
+they are at rest relative to it. We now ask ourselves whether both clocks go at
+the same rate from the standpoint of the non-rotating Galileian reference-body
+<i>K</i>. As judged from this body, the clock at the centre of the disc has no
+velocity, whereas the clock at the edge of the disc is in motion relative to <i>K</i>
+in consequence of the rotation. According to a result obtained in Section XII,
+it follows that the latter clock goes at a rate permanently slower than that of
+the clock at the centre of the circular disc, <i>i.e.</i> as observed from <i>K</i>. It is
+obvious that the same effect would be noted by an observer whom we will imagine
+sitting alongside his clock at the centre of the circular disc. Thus on our
+circular disc, or, to make the case more general, in every gravitational field,
+a clock will go more quickly or less quickly, according to the position in
+which the clock is situated (at rest). For this reason it is not possible to
+obtain a reasonable definition of time with the aid of clocks which are
+arranged at rest with respect to the body of reference. A similar difficulty
+presents itself when we attempt to apply our earlier definition of simultaneity
+in such a case, but I do not wish to go any farther into this question.
+</p>
+
+<p>
+Moreover, at this stage the definition of the space co-ordinates also presents
+insurmountable difficulties. If the observer applies his standard measuring-rod
+(a rod which is short as compared with the radius of the disc) tangentially to
+the edge of the disc, then, as judged from the Galileian system, the length of
+this rod will be less than 1, since, according to Section XII, moving bodies
+suffer a shortening in the direction of the motion. On the other hand, the
+measuring-rod will not experience a shortening in length, as judged from <i>K</i>, if
+it is applied to the disc in the direction of the radius. If, then, the
+observer first measures the circumference of the disc with his measuring-rod
+and then the diameter of the disc, on dividing the one by the other, he will
+not obtain as quotient the familiar number &#960; = 3.14 . . ., but a larger
+number,<a href="#linknote-19" name="linknoteref-19" id="linknoteref-19">[19]</a> whereas of course, for a disc which is at rest with
+respect to <i>K</i>, this operation would yield &#960; exactly. This proves that the
+propositions of Euclidean geometry cannot hold exactly on the rotating disc,
+nor in general in a gravitational field, at least if we attribute the length 1
+to the rod in all positions and in every orientation. Hence the idea of a
+straight line also loses its meaning. We are therefore not in a position to
+define exactly the co-ordinates <i>x, y, z</i> relative to the disc by means of the
+method used in discussing the special theory, and as long as the co-ordinates
+and times of events have not been defined, we cannot assign an exact meaning to
+the natural laws in which these occur.
+</p>
+
+<p>
+<a name="linknote-19" id="linknote-19">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-19"> [19]</a><br/> Throughout this consideration we have
+to use the Galileian (non-rotating) system <i>K</i> as reference-body, since we
+may only assume the validity of the results of the special theory of relativity
+relative to <i>K</i> (relative to <i>K&#x2032;</i> a gravitational field prevails).
+</p>
+
+<p>
+Thus all our previous conclusions based on general relativity would appear to
+be called in question. In reality we must make a subtle detour in order to be
+able to apply the postulate of general relativity exactly. I shall prepare the
+reader for this in the following paragraphs.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap24"></a>XXIV.<br/>
+EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM</h3>
+
+<p>
+The surface of a marble table is spread out in front of me. I can get from any
+one point on this table to any other point by passing continuously from one
+point to a “neighbouring” one, and repeating this process a (large)
+number of times, or, in other words, by going from point to point without
+executing “jumps.” I am sure the reader will appreciate with
+sufficient clearness what I mean here by “neighbouring” and by
+“jumps” (if he is not too pedantic). We express this property of the
+surface by describing the latter as a continuum.
+</p>
+
+<p>
+Let us now imagine that a large number of little rods of equal length have been
+made, their lengths being small compared with the dimensions of the marble
+slab. When I say they are of equal length, I mean that one can be laid on any
+other without the ends overlapping. We next lay four of these little rods on
+the marble slab so that they constitute a quadrilateral figure (a square), the
+diagonals of which are equally long. To ensure the equality of the diagonals,
+we make use of a little testing-rod. To this square we add similar ones, each
+of which has one rod in common with the first. We proceed in like manner with
+each of these squares until finally the whole marble slab is laid out with
+squares. The arrangement is such, that each side of a square belongs to two
+squares and each corner to four squares.
+</p>
+
+<p>
+It is a veritable wonder that we can carry out this business without getting
+into the greatest difficulties. We only need to think of the following. If at
+any moment three squares meet at a corner, then two sides of the fourth square
+are already laid, and, as a consequence, the arrangement of the remaining two
+sides of the square is already completely determined. But I am now no longer
+able to adjust the quadrilateral so that its diagonals may be equal. If they
+are equal of their own accord, then this is an especial favour of the marble
+slab and of the little rods, about which I can only be thankfully surprised. We
+must experience many such surprises if the construction is to be successful.
+</p>
+
+<p>
+If everything has really gone smoothly, then I say that the points of the
+marble slab constitute a Euclidean continuum with respect to the little rod,
+which has been used as a “distance” (line-interval). By choosing one
+corner of a square as “origin” I can characterise every other corner
+of a square with reference to this origin by means of two numbers. I only need
+state how many rods I must pass over when, starting from the origin, I proceed
+towards the “right” and then “upwards,” in order to arrive
+at the corner of the square under consideration. These two numbers are then the
+“Cartesian co-ordinates” of this corner with reference to the
+“Cartesian co-ordinate system” which is determined by the arrangement
+of little rods.
+</p>
+
+<p>
+By making use of the following modification of this abstract experiment, we
+recognise that there must also be cases in which the experiment would be
+unsuccessful. We shall suppose that the rods “expand” by in amount
+proportional to the increase of temperature. We heat the central part of the
+marble slab, but not the periphery, in which case two of our little rods can
+still be brought into coincidence at every position on the table. But our
+construction of squares must necessarily come into disorder during the heating,
+because the little rods on the central region of the table expand, whereas
+those on the outer part do not.
+</p>
+
+<p>
+With reference to our little rods—defined as unit lengths—the marble slab is
+no longer a Euclidean continuum, and we are also no longer in the position of
+defining Cartesian co-ordinates directly with their aid, since the above
+construction can no longer be carried out. But since there are other things
+which are not influenced in a similar manner to the little rods (or perhaps not
+at all) by the temperature of the table, it is possible quite naturally to
+maintain the point of view that the marble slab is a “Euclidean
+continuum.” This can be done in a satisfactory manner by making a more
+subtle stipulation about the measurement or the comparison of lengths.
+</p>
+
+<p>
+But if rods of every kind (<i>i.e.</i> of every material) were to behave <i>in
+the same way</i> as regards the influence of temperature when they are on the
+variably heated marble slab, and if we had no other means of detecting the
+effect of temperature than the geometrical behaviour of our rods in experiments
+analogous to the one described above, then our best plan would be to assign the
+distance one to two points on the slab, provided that the ends of one of our
+rods could be made to coincide with these two points; for how else should we
+define the distance without our proceeding being in the highest measure grossly
+arbitrary? The method of Cartesian coordinates must then be discarded, and
+replaced by another which does not assume the validity of Euclidean geometry
+for rigid bodies.<a href="#linknote-20" name="linknoteref-20"
+id="linknoteref-20">[20]</a> The reader will notice that the situation depicted
+here corresponds to the one brought about by the general postulate of
+relativity (Section XXIII).
+</p>
+
+<p>
+<a name="linknote-20" id="linknote-20">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-20"> [20]</a><br/> Mathematicians have been confronted
+with our problem in the following form. If we are given a surface (<i>e.g.</i> an
+ellipsoid) in Euclidean three-dimensional space, then there exists for this
+surface a two-dimensional geometry, just as much as for a plane surface. Gauss
+undertook the task of treating this two-dimensional geometry from first
+principles, without making use of the fact that the surface belongs to a
+Euclidean continuum of three dimensions. If we imagine constructions to be made
+with rigid rods <i>in the surface</i> (similar to that above with the marble
+slab), we should find that different laws hold for these from those resulting
+on the basis of Euclidean plane geometry. The surface is not a Euclidean
+continuum with respect to the rods, and we cannot define Cartesian co-ordinates
+<i>in the surface</i>. Gauss indicated the principles according to which we can
+treat the geometrical relationships in the surface, and thus pointed out the
+way to the method of Riemann of treating multi-dimensional, non-Euclidean
+<i>continuum</i>. Thus it is that mathematicians long ago solved the formal
+problems to which we are led by the general postulate of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap25"></a>XXV.<br/>
+GAUSSIAN CO-ORDINATES</h3>
+
+<div class="fig" style="width:50%;">
+<img src="images/image033.jpg" style="width:100%;" alt="image033" /><br/><br/>
+</div>
+
+<p>
+According to Gauss, this combined analytical and geometrical mode of handling
+the problem can be arrived at in the following way. We imagine a system of
+arbitrary curves (see Fig. 4) drawn on the surface of the table. These we
+designate as <i>u</i>-curves, and we indicate each of them by means of a number. The
+Curves <i>u</i> = 1, <i>u</i> = 2 and <i>u</i> = 3 are drawn in the diagram. Between the curves <i>u</i> =
+1 and <i>u</i> = 2 we must imagine an infinitely large number to be drawn, all of
+which correspond to real numbers lying between 1 and 2. We have then a
+system of <i>u</i>-curves, and this “infinitely dense” system covers the
+whole surface of the table. These <i>u</i>-curves must not intersect each other, and
+through each point of the surface one and only one curve must pass. Thus a
+perfectly definite value of <i>u</i> belongs to every point on the surface of the
+marble slab. In like manner we imagine a system of <i>v</i>-curves drawn on the
+surface. These satisfy the same conditions as the <i>u</i>-curves, they are provided
+with numbers in a corresponding manner, and they may likewise be of arbitrary
+shape. It follows that a value of <i>u</i> and a value of <i>v</i> belong to every point on
+the surface of the table. We call these two numbers the co-ordinates of the
+surface of the table (Gaussian co-ordinates). For example, the point <i>P</i> in the
+diagram has the Gaussian co-ordinates <i>u</i> = 3, <i>v</i> = 1. Two neighbouring points <i>P</i>
+and <i>P&#x2032;</i> on the surface then correspond to the co-ordinates
+</p>
+
+<p>
+<i>P</i>: <i>u, v</i>
+</p>
+
+<p>
+<i>P&#x2032;</i>: <i>u</i> + <i>du, v</i> + <i>dv</i>,
+</p>
+
+<p class="noindent">
+where <i>du</i> and <i>dv</i> signify very small numbers. In a similar manner we may indicate
+the distance (line-interval) between <i>P</i> and <i>P&#x2032;</i>, as measured with a
+little rod, by means of the very small number <i>ds</i>. Then according to Gauss we
+have
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>g</i><sub>11</sub><i>du</i><sup>2</sup> + 2<i>g</i><sub>12</sub><i>du dv</i> +
+<i>g</i><sub>22</sub><i>dv</i><sup>2</sup>,
+</p>
+
+<p class="noindent">
+where <i>g</i><sub>11</sub>, <i>g</i><sub>12</sub>, <i>g</i><sub>22</sub>, are magnitudes which
+depend in a perfectly definite way on <i>u</i> and <i>v</i>. The magnitudes <i>g</i><sub>11</sub>,
+<i>g</i><sub>12</sub> and <i>g</i><sub>22</sub>, determine the behaviour of the rods relative
+to the <i>u</i>-curves and <i>v</i>-curves, and thus also relative to the surface of the
+table. For the case in which the points of the surface considered form a
+Euclidean continuum with reference to the measuring-rods, but only in this
+case, it is possible to draw the <i>u</i>-curves and <i>v</i>-curves and to attach numbers to
+them, in such a manner, that we simply have:
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>du</i><sup>2</sup> + <i>dv</i><sup>2</sup>
+</p>
+
+<p class="noindent">
+Under these conditions, the <i>u</i>-curves and <i>v</i>-curves are straight lines in the
+sense of Euclidean geometry, and they are perpendicular to each other. Here the
+Gaussian coordinates are simply Cartesian ones. It is clear that Gauss
+co-ordinates are nothing more than an association of two sets of numbers with
+the points of the surface considered, of such a nature that numerical values
+differing very slightly from each other are associated with neighbouring points
+“in space.”
+</p>
+
+<p>
+So far, these considerations hold for a continuum of two dimensions. But the
+Gaussian method can be applied also to a continuum of three, four or more
+dimensions. If, for instance, a continuum of four dimensions be supposed
+available, we may represent it in the following way. With every point of the
+continuum, we associate arbitrarily four numbers, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,
+<i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, which are known as “co-ordinates.”
+Adjacent points correspond to adjacent values of the coordinates. If a distance
+<i>ds</i> is associated with the adjacent points <i>P</i> and <i>P&#x2032;</i>, this distance
+being measurable and well defined from a physical point of view, then the
+following formula holds:
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>g</i><sub>11</sub><i>dx</i><sub>1</sub><sup>2</sup>
++ 2<i>g</i><sub>12</sub><i>dx</i><sub>1</sub><i>dx</i><sub>2</sub> . . . . +
+<i>g</i><sub>44</sub><i>dx</i><sub>4</sub><sup>2</sup>,
+</p>
+
+<p class="noindent">
+where the magnitudes <i>g</i><sub>11</sub>, etc., have values which vary with the
+position in the continuum. Only when the continuum is a Euclidean one is it
+possible to associate the co-ordinates <i>x</i><sub>1</sub> . . <i>x</i><sub>4</sub>. with
+the points of the continuum so that we have simply
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>dx</i><sub>1</sub><sup>2</sup> +
+<i>dx</i><sub>2</sub><sup>2</sup> + <i>dx</i><sub>3</sub><sup>2</sup> +
+<i>dx</i><sub>4</sub><sup>2</sup>.
+</p>
+
+<p class="noindent">
+In this case relations hold in the four-dimensional continuum which are
+analogous to those holding in our three-dimensional measurements.
+</p>
+
+<p>
+However, the Gauss treatment for <i>ds</i><sup>2</sup> which we have given above is
+not always possible. It is only possible when sufficiently small regions of the
+continuum under consideration may be regarded as Euclidean continua. For
+example, this obviously holds in the case of the marble slab of the table and
+local variation of temperature. The temperature is practically constant for a
+small part of the slab, and thus the geometrical behaviour of the rods is
+<i>almost</i> as it ought to be according to the rules of Euclidean geometry. Hence
+the imperfections of the construction of squares in the previous section do not
+show themselves clearly until this construction is extended over a considerable
+portion of the surface of the table.
+</p>
+
+<p>
+We can sum this up as follows: Gauss invented a method for the mathematical
+treatment of continua in general, in which “size-relations”
+(“distances” between neighbouring points) are defined. To every point
+of a continuum are assigned as many numbers (Gaussian coordinates) as the
+continuum has dimensions. This is done in such a way, that only one meaning can
+be attached to the assignment, and that numbers (Gaussian coordinates) which
+differ by an indefinitely small amount are assigned to adjacent points. The
+Gaussian coordinate system is a logical generalisation of the Cartesian
+co-ordinate system. It is also applicable to non-Euclidean continua, but only
+when, with respect to the defined “size” or “distance,”
+small parts of the continuum under consideration behave more nearly like a
+Euclidean system, the smaller the part of the continuum under our notice.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap26"></a>XXVI.<br/>
+THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED AS A
+EUCLIDEAN CONTINUUM</h3>
+
+<p>
+We are now in a position to formulate more exactly the idea of Minkowski, which
+was only vaguely indicated in Section XVII. In accordance with the special theory
+of relativity, certain co-ordinate systems are given preference for the
+description of the four-dimensional, space-time continuum. We called these
+“Galileian co-ordinate systems.” For these systems, the four
+co-ordinates <i>x, y, z, t</i>, which determine an event or—in other words—a point
+of the four-dimensional continuum, are defined physically in a simple manner, as
+set forth in detail in the first part of this book. For the transition from one
+Galileian system to another, which is moving uniformly with reference to the
+first, the equations of the Lorentz transformation are valid. These last form
+the basis for the derivation of deductions from the special theory of
+relativity, and in themselves they are nothing more than the expression of the
+universal validity of the law of transmission of light for all Galileian
+systems of reference.
+</p>
+
+<p>
+Minkowski found that the Lorentz transformations satisfy the following simple
+conditions. Let us consider two neighbouring events, the relative position of
+which in the four-dimensional continuum is given with respect to a Galileian
+reference-body <i>K</i> by the space co-ordinate differences <i>dx, dy, dz</i>
+and the time-difference <i>dt</i>. With reference to a second Galileian system
+we shall suppose that the corresponding differences for these two events are
+<i>dx&#x2032;, dy&#x2032;, dz&#x2032;, dt&#x2032;</i>. Then these magnitudes always fulfill the condition.<a href="#linknote-21" name="linknoteref-21" id="linknoteref-21">[21]</a>
+</p>
+
+<p>
+<a name="linknote-21" id="linknote-21">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-21"> [21]</a><br/> Cf. Appendixes I and II. The relations
+which are derived there for the co-ordinates themselves are valid also for
+co-ordinate <i>differences</i>, and thus also for co-ordinate differentials
+(indefinitely small differences).
+</p>
+
+<p class="center">
+<i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>dt</i><sup>2</sup> = <i>dx&#x2032;</i><sup>2</sup> +
+<i>dy&#x2032;</i><sup>2</sup> + <i>dz&#x2032;</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>dt&#x2032;</i><sup>2</sup>.
+</p>
+
+<p>
+The validity of the Lorentz transformation follows from this condition. We can
+express this as follows: The magnitude
+</p>
+
+<p class="center">
+<i>ds</i><sup>2</sup> = <i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup> –
+<i>c</i><sup>2</sup> <i>dt</i><sup>2</sup>,
+</p>
+
+<p class="noindent">
+which belongs to two adjacent points of the four-dimensional space-time
+continuum, has the same value for all selected (Galileian) reference-bodies. If
+we replace <i>x, y, z</i>,
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image034.jpg" style="width:100%;" alt="image034" /><br/><br/>
+</div>
+
+<p class="noindent">
+by <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, we also obtain
+the result that
+</p>
+
+<p class="center">
+<i>ds</i><sup>2</sup> = <i>dx</i><sub>1</sub><sup>2</sup> + <i>dx</i><sub>2</sub><sup>2</sup> +
+<i>dx</i><sub>3</sub><sup>2</sup> + <i>dx</i><sub>4</sub><sup>2</sup>.
+</p>
+
+<p class="noindent">
+is independent of the choice of the body of reference. We call the magnitude <i>ds</i>
+the “distance” apart of the two events or four-dimensional points.
+</p>
+
+<p>
+Thus, if we choose as time-variable the imaginary variable
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image035.jpg" style="width:100%;" alt="image035" /><br/><br/>
+</div>
+
+<p class="noindent">
+instead of the real quantity <i>t</i>, we can regard the space-time
+contintium—accordance with the special theory of relativity—as a
+“Euclidean” four-dimensional continuum, a result which follows from
+the considerations of the preceding section.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap27"></a>XXVII.<br/>
+THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A EUCLIDEAN
+CONTINUUM</h3>
+
+<p>
+In the first part of this book we were able to make use of space-time
+co-ordinates which allowed of a simple and direct physical interpretation, and
+which, according to Section XXVI, can be regarded as four-dimensional Cartesian
+co-ordinates. This was possible on the basis of the law of the constancy of the
+velocity of light. But according to Section XXI the general theory of relativity
+cannot retain this law. On the contrary, we arrived at the result that
+according to this latter theory the velocity of light must always depend on the
+co-ordinates when a gravitational field is present. In connection with a
+specific illustration in Section XXIII, we found that the presence of a
+gravitational field invalidates the definition of the coordinates and the time,
+which led us to our objective in the special theory of relativity.
+</p>
+
+<p>
+In view of the resuIts of these considerations we are led to the conviction
+that, according to the general principle of relativity, the space-time
+continuum cannot be regarded as a Euclidean one, but that here we have the
+general case, corresponding to the marble slab with local variations of
+temperature, and with which we made acquaintance as an example of a
+two-dimensional continuum. Just as it was there impossible to construct a
+Cartesian co-ordinate system from equal rods, so here it is impossible to build
+up a system (reference-body) from rigid bodies and clocks, which shall be of
+such a nature that measuring-rods and clocks, arranged rigidly with respect to
+one another, shall indicate position and time directly. Such was the essence of
+the difficulty with which we were confronted in Section XXIII.
+</p>
+
+<p>
+But the considerations of Sections XXV and XXVI show us the way to surmount this
+difficulty. We refer the four-dimensional space-time continuum in an arbitrary
+manner to Gauss co-ordinates. We assign to every point of the continuum (event)
+four numbers, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>
+(co-ordinates), which have not the least direct physical significance, but only
+serve the purpose of numbering the points of the continuum in a definite but
+arbitrary manner. This arrangement does not even need to be of such a kind that
+we must regard <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, as
+“space” co-ordinates and <i>x</i><sub>4</sub>, as a “time”
+co-ordinate.
+</p>
+
+<p>
+The reader may think that such a description of the world would be quite
+inadequate. What does it mean to assign to an event the particular co-ordinates
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, if in themselves
+these co-ordinates have no significance? More careful consideration shows,
+however, that this anxiety is unfounded. Let us consider, for instance, a
+material point with any kind of motion. If this point had only a momentary
+existence without duration, then it would to described in space-time by a
+single system of values <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>,
+<i>x</i><sub>4</sub>. Thus its permanent existence must be characterised by an
+infinitely large number of such systems of values, the co-ordinate values of
+which are so close together as to give continuity; corresponding to the
+material point, we thus have a (uni-dimensional) line in the four-dimensional
+continuum. In the same way, any such lines in our continuum correspond to many
+points in motion. The only statements having regard to these points which can
+claim a physical existence are in reality the statements about their
+encounters. In our mathematical treatment, such an encounter is expressed in
+the fact that the two lines which represent the motions of the points in
+question have a particular system of co-ordinate values, <i>x</i><sub>1</sub>,
+<i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, in common. After mature
+consideration the reader will doubtless admit that in reality such encounters
+constitute the only actual evidence of a time-space nature with which we meet
+in physical statements.
+</p>
+
+<p>
+When we were describing the motion of a material point relative to a body of
+reference, we stated nothing more than the encounters of this point with
+particular points of the reference-body. We can also determine the
+corresponding values of the time by the observation of encounters of the body
+with clocks, in conjunction with the observation of the encounter of the hands
+of clocks with particular points on the dials. It is just the same in the case
+of space-measurements by means of measuring-rods, as a little consideration
+will show.
+</p>
+
+<p>
+The following statements hold generally: Every physical description resolves
+itself into a number of statements, each of which refers to the space-time
+coincidence of two events <i>A</i> and <i>B</i>. In terms of Gaussian co-ordinates, every
+such statement is expressed by the agreement of their four co-ordinates
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>. Thus in reality,
+the description of the time-space continuum by means of Gauss co-ordinates
+completely replaces the description with the aid of a body of reference,
+without suffering from the defects of the latter mode of description; it is not
+tied down to the Euclidean character of the continuum which has to be
+represented.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap28"></a>XXVIII.<br/>
+EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+We are now in a position to replace the provisional formulation of the general
+principle of relativity given in Section XVIII by an exact formulation. The form
+there used, “All bodies of reference <i>K, K&#x2032;</i>, etc., are equivalent for the
+description of natural phenomena (formulation of the general laws of nature),
+whatever may be their state of motion,” cannot be maintained, because the
+use of rigid reference-bodies, in the sense of the method followed in the
+special theory of relativity, is in general not possible in space-time
+description. The Gauss co-ordinate system has to take the place of the body of
+reference. The following statement corresponds to the fundamental idea of the
+general principle of relativity: “<i>All Gaussian co-ordinate systems are
+essentially equivalent for the formulation of the general laws of nature.</i>”
+</p>
+
+<p>
+We can state this general principle of relativity in still another form, which
+renders it yet more clearly intelligible than it is when in the form of the
+natural extension of the special principle of relativity. According to the
+special theory of relativity, the equations which express the general laws of
+nature pass over into equations of the same form when, by making use of the
+Lorentz transformation, we replace the space-time variables <i>x, y, z, t</i>, of a
+(Galileian) reference-body <i>K</i> by the space-time variables <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>, of a
+new reference-body <i>K&#x2032;</i>. According to the general theory of relativity, on the
+other hand, by application of <i>arbitrary substitutions</i> of the Gauss variables
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, the equations must
+pass over into equations of the same form; for every transformation (not only
+the Lorentz transformation) corresponds to the transition of one Gauss
+co-ordinate system into another.
+</p>
+
+<p>
+If we desire to adhere to our “old-time” three-dimensional view of
+things, then we can characterise the development which is being undergone by
+the fundamental idea of the general theory of relativity as follows: The
+special theory of relativity has reference to Galileian domains, <i>i.e.</i> to those
+in which no gravitational field exists. In this connection a Galileian
+reference-body serves as body of reference, <i>i.e.</i> a rigid body the state of
+motion of which is so chosen that the Galileian law of the uniform rectilinear
+motion of “isolated” material points holds relatively to it.
+</p>
+
+<p>
+Certain considerations suggest that we should refer the same Galileian domains
+to <i>non-Galileian</i> reference-bodies also. A gravitational field of a special kind
+is then present with respect to these bodies (cf. Sections XX and XXIII).
+</p>
+
+<p>
+In gravitational fields there are no such things as rigid bodies with Euclidean
+properties; thus the fictitious rigid body of reference is of no avail in the
+general theory of relativity. The motion of clocks is also influenced by
+gravitational fields, and in such a way that a physical definition of time
+which is made directly with the aid of clocks has by no means the same degree
+of plausibility as in the special theory of relativity.
+</p>
+
+<p>
+For this reason non-rigid reference-bodies are used, which are as a whole not
+only moving in any way whatsoever, but which also suffer alterations in form <i>ad
+lib.</i> during their motion. Clocks, for which the law of motion is of any kind,
+however irregular, serve for the definition of time. We have to imagine each of
+these clocks fixed at a point on the non-rigid reference-body. These clocks
+satisfy only the one condition, that the “readings” which are
+observed simultaneously on adjacent clocks (in space) differ from each other by
+an indefinitely small amount. This non-rigid reference-body, which might
+appropriately be termed a “reference-mollusc”, is in the main
+equivalent to a Gaussian four-dimensional co-ordinate system chosen
+arbitrarily. That which gives the “mollusc” a certain
+comprehensibility as compared with the Gauss co-ordinate system is the (really
+unjustified) formal retention of the separate existence of the
+</p>
+
+<p>
+space co-ordinates as opposed to the time co-ordinate. Every point on the
+mollusc is treated as a space-point, and every material point which is at rest
+relatively to it as at rest, so long as the mollusc is considered as
+reference-body. The general principle of relativity requires that all these
+molluscs can be used as reference-bodies with equal right and equal success in
+the formulation of the general laws of nature; the laws themselves must be
+quite independent of the choice of mollusc.
+</p>
+
+<p>
+The great power possessed by the general principle of relativity lies in the
+comprehensive limitation which is imposed on the laws of nature in consequence
+of what we have seen above.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap29"></a>XXIX.<br/>
+THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
+PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+If the reader has followed all our previous considerations, he will have no
+further difficulty in understanding the methods leading to the solution of the
+problem of gravitation.
+</p>
+
+<p>
+We start off on a consideration of a Galileian domain, <i>i.e.</i> a domain in which
+there is no gravitational field relative to the Galileian reference-body <i>K</i>. The
+behaviour of measuring-rods and clocks with reference to <i>K</i> is known from the
+special theory of relativity, likewise the behaviour of “isolated”
+material points; the latter move uniformly and in straight lines.
+</p>
+
+<p>
+Now let us refer this domain to a random Gauss coordinate system or to a
+“mollusc” as reference-body <i>K&#x2032;</i>. Then with respect to <i>K&#x2032;</i> there is a
+gravitational field <i>G</i> (of a particular kind). We learn the behaviour of
+measuring-rods and clocks and also of freely-moving material points with
+reference to <i>K&#x2032;</i> simply by mathematical transformation. We interpret this
+behaviour as the behaviour of measuring-rods, clocks and material points under
+the influence of the gravitational field <i>G</i>. Hereupon we introduce a hypothesis:
+that the influence of the gravitational field on measuring-rods, clocks and
+freely-moving material points continues to take place according to the same
+laws, even in the case where the prevailing gravitational field is <i>not</i>
+derivable from the Galileian special case, simply by means of a transformation
+of co-ordinates.
+</p>
+
+<p>
+The next step is to investigate the space-time behaviour of the gravitational
+field <i>G</i>, which was derived from the Galileian special case simply by
+transformation of the coordinates. This behaviour is formulated in a law, which
+is always valid, no matter how the reference-body (mollusc) used in the
+description may be chosen.
+</p>
+
+<p>
+This law is not yet the <i>general</i> law of the gravitational field, since the
+gravitational field under consideration is of a special kind. In order to find
+out the general law-of-field of gravitation we still require to obtain a
+generalisation of the law as found above. This can be obtained without caprice,
+however, by taking into consideration the following demands:
+</p>
+
+<p class="letter">
+(<i>a</i>) The required generalisation must likewise satisfy the general postulate of
+relativity.
+</p>
+
+<p class="letter">
+(<i>b</i>) If there is any matter in the domain under consideration, only its inertial
+mass, and thus according to Section XV only its energy is of importance for its
+effect in exciting a field.
+</p>
+
+<p class="letter">
+(<i>c</i>) Gravitational field and matter together must satisfy the law of the
+conservation of energy (and of impulse).
+</p>
+
+<p>
+Finally, the general principle of relativity permits us to determine the
+influence of the gravitational field on the course of all those processes which
+take place according to known laws when a gravitational field is absent <i>i.e.</i>
+which have already been fitted into the frame of the special theory of
+relativity. In this connection we proceed in principle according to the method
+which has already been explained for measuring-rods, clocks and freely moving
+material points.
+</p>
+
+<p>
+The theory of gravitation derived in this way from the general postulate of
+relativity excels not only in its beauty; nor in removing the defect attaching
+to classical mechanics which was brought to light in Section XXI; nor in
+interpreting the empirical law of the equality of inertial and gravitational
+mass; but it has also already explained a result of observation in astronomy,
+against which classical mechanics is powerless.
+</p>
+
+<p>
+If we confine the application of the theory to the case where the gravitational
+fields can be regarded as being weak, and in which all masses move with respect
+to the coordinate system with velocities which are small compared with the
+velocity of light, we then obtain as a first approximation the Newtonian
+theory. Thus the latter theory is obtained here without any particular
+assumption, whereas Newton had to introduce the hypothesis that the force of
+attraction between mutually attracting material points is inversely
+proportional to the square of the distance between them. If we increase the
+accuracy of the calculation, deviations from the theory of Newton make their
+appearance, practically all of which must nevertheless escape the test of
+observation owing to their smallness.
+</p>
+
+<p>
+We must draw attention here to one of these deviations. According to Newton’s
+theory, a planet moves round the sun in an ellipse, which would permanently
+maintain its position with respect to the fixed stars, if we could disregard
+the motion of the fixed stars themselves and the action of the other planets
+under consideration. Thus, if we correct the observed motion of the planets for
+these two influences, and if Newton’s theory be strictly correct, we ought to
+obtain for the orbit of the planet an ellipse, which is fixed with reference to
+the fixed stars. This deduction, which can be tested with great accuracy, has
+been confirmed for all the planets save one, with the precision that is capable
+of being obtained by the delicacy of observation attainable at the present
+time. The sole exception is Mercury, the planet which lies nearest the sun.
+Since the time of Leverrier, it has been known that the ellipse corresponding
+to the orbit of Mercury, after it has been corrected for the influences
+mentioned above, is not stationary with respect to the fixed stars, but that it
+rotates exceedingly slowly in the plane of the orbit and in the sense of the
+orbital motion. The value obtained for this rotary movement of the orbital
+ellipse was 43 seconds of arc per century, an amount ensured to be correct to
+within a few seconds of arc. This effect can be explained by means of classical
+mechanics only on the assumption of hypotheses which have little probability,
+and which were devised solely for this purponse.
+</p>
+
+<p>
+On the basis of the general theory of relativity, it is found that the ellipse
+of every planet round the sun must necessarily rotate in the manner indicated
+above; that for all the planets, with the exception of Mercury, this rotation
+is too small to be detected with the delicacy of observation possible at the
+present time; but that in the case of Mercury it must amount to 43 seconds of
+arc per century, a result which is strictly in agreement with observation.
+</p>
+
+<p>
+Apart from this one, it has hitherto been possible to make only two deductions
+from the theory which admit of being tested by observation, to wit, the
+curvature of light rays by the gravitational field of the sun,<a href="#linknote-22" name="linknoteref-22" id="linknoteref-22">[22]</a>
+and a displacement of the spectral lines of light reaching us from large stars,
+as compared with the corresponding lines for light produced in an analogous
+manner terrestrially (<i>i.e.</i> by the same kind of atom).<a href="#linknote-23" name="linknoteref-23" id="linknoteref-23">[23]</a> These two
+deductions from the theory have both been confirmed.
+</p>
+
+<p>
+<a name="linknote-22" id="linknote-22">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-22"> [22]</a><br/> First observed by Eddington and others
+in 1919. (Cf. Appendix III).
+</p>
+
+<p>
+<a name="linknote-23" id="linknote-23">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-23"> [23]</a><br/> Established by Adams in 1924. (Cf. p.
+132)
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part03"></a>PART III: CONSIDERATIONS ON THE UNIVERSE AS A
+WHOLE</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap30"></a>XXX.<br/>
+COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY</h3>
+
+<p>
+Part from the difficulty discussed in Section XXI, there is a second fundamental
+difficulty attending classical celestial mechanics, which, to the best of my
+knowledge, was first discussed in detail by the astronomer Seeliger. If we
+ponder over the question as to how the universe, considered as a whole, is to
+be regarded, the first answer that suggests itself to us is surely this: As
+regards space (and time) the universe is infinite. There are stars everywhere,
+so that the density of matter, although very variable in detail, is
+nevertheless on the average everywhere the same. In other words: However far we
+might travel through space, we should find everywhere an attenuated swarm of
+fixed stars of approrimately the same kind and density.
+</p>
+
+<p>
+This view is not in harmony with the theory of Newton. The latter theory rather
+requires that the universe should have a kind of centre in which the density of
+the stars is a maximum, and that as we proceed outwards from this centre the
+group-density of the stars should diminish, until finally, at great distances,
+it is succeeded by an infinite region of emptiness. The stellar universe ought
+to be a finite island in the infinite ocean of space.<a href="#linknote-24" name="linknoteref-24" id="linknoteref-24">[24]</a>
+</p>
+
+<p>
+<a name="linknote-24" id="linknote-24">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-24"> [24]</a><br/> <i>Proof</i>—According to the
+theory of Newton, the number of “lines of force” which come from
+infinity and terminate in a mass m is proportional to the mass <i>m</i>. If, on
+the average, the mass density &#961;<sub>0 </sub>is constant throughout the
+universe, then a sphere of volume <i>V</i> will enclose the average mass
+&#961;<sub>0</sub><i>V</i>. Thus the number of lines of force passing through
+the surface <i>F</i> of the sphere into its interior is proportional to
+&#961;<sub>0</sub><i>V</i>. For unit area of the surface of the sphere the
+number of lines of force which enters the sphere is thus proportional to
+&#961;<sub>0</sub><i>V/F</i> or to &#961;<sub>0</sub><i>R</i>. Hence the
+intensity of the field at the surface would ultimately become infinite with
+increasing radius <i>R</i> of the sphere, which is impossible.
+</p>
+
+<p>
+This conception is in itself not very satisfactory. It is still less
+satisfactory because it leads to the result that the light emitted by the stars
+and also individual stars of the stellar system are perpetually passing out
+into infinite space, never to return, and without ever again coming into
+interaction with other objects of nature. Such a finite material universe would
+be destined to become gradually but systematically impoverished.
+</p>
+
+<p>
+In order to escape this dilemma, Seeliger suggested a modification of Newton’s
+law, in which he assumes that for great distances the force of attraction
+between two masses diminishes more rapidly than would result from the inverse
+square law. In this way it is possible for the mean density of matter to be
+constant everywhere, even to infinity, without infinitely large gravitational
+fields being produced. We thus free ourselves from the distasteful conception
+that the material universe ought to possess something of the nature of a
+centre. Of course we purchase our emancipation from the fundamental
+difficulties mentioned, at the cost of a modification and complication of
+Newton’s law which has neither empirical nor theoretical foundation. We can
+imagine innumerable laws which would serve the same purpose, without our being
+able to state a reason why one of them is to be preferred to the others; for
+any one of these laws would be founded just as little on more general
+theoretical principles as is the law of Newton.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap31"></a>XXXI.<br/>
+THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED”
+UNIVERSE</h3>
+
+<p>
+But speculations on the structure of the universe also move in quite another
+direction. The development of non-Euclidean geometry led to the recognition of
+the fact, that we can cast doubt on the <i>infiniteness</i> of our space without
+coming into conflict with the laws of thought or with experience (Riemann,
+Helmholtz). These questions have already been treated in detail and with
+unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on
+them briefly here.
+</p>
+
+<p>
+In the first place, we imagine an existence in two dimensional space. Flat
+beings with flat implements, and in particular flat rigid measuring-rods, are
+free to move in a <i>plane</i>. For them nothing exists outside of this plane: that
+which they observe to happen to themselves and to their flat “things”
+is the all-inclusive reality of their plane. In particular, the constructions
+of plane Euclidean geometry can be carried out by means of the rods <i>e.g.</i> the
+lattice construction, considered in Section XXIV. In contrast to ours, the
+universe of these beings is two-dimensional; but, like ours, it extends to
+infinity. In their universe there is room for an infinite number of identical
+squares made up of rods, <i>i.e.</i> its volume (surface) is infinite. If these beings
+say their universe is “plane,” there is sense in the statement,
+because they mean that they can perform the constructions of plane Euclidean
+geometry with their rods. In this connection the individual rods always
+represent the same distance, independently of their position.
+</p>
+
+<p>
+Let us consider now a second two-dimensional existence, but this time on a
+spherical surface instead of on a plane. The flat beings with their
+measuring-rods and other objects fit exactly on this surface and they are
+unable to leave it. Their whole universe of observation extends exclusively
+over the surface of the sphere. Are these beings able to regard the geometry of
+their universe as being plane geometry and their rods withal as the realisation
+of “distance”? They cannot do this. For if they attempt to realise a
+straight line, they will obtain a curve, which we “three-dimensional
+beings” designate as a great circle, <i>i.e.</i> a self-contained line of
+definite finite length, which can be measured up by means of a measuring-rod.
+Similarly, this universe has a finite area that can be compared with the area,
+of a square constructed with rods. The great charm resulting from this
+consideration lies in the recognition of the fact that <i>the universe of these
+beings is finite and yet has no limits.</i>
+</p>
+
+<p>
+But the spherical-surface beings do not need to go on a world-tour in order to
+perceive that they are not living in a Euclidean universe. They can convince
+themselves of this on every part of their “world,” provided they do
+not use too small a piece of it. Starting from a point, they draw
+“straight lines” (arcs of circles as judged in three dimensional
+space) of equal length in all directions. They will call the line joining the
+free ends of these lines a “circle.” For a plane surface, the ratio
+of the circumference of a circle to its diameter, both lengths being measured
+with the same rod, is, according to Euclidean geometry of the plane, equal to a
+constant value &#960;, which is independent of the diameter of the circle. On
+their spherical surface our flat beings would find for this ratio the value
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image036.jpg" style="width:100%;" alt="image036" /><br/><br/>
+</div>
+
+<p class="noindent">
+<i>i.e.</i> a smaller value than &#960;, the difference being the more considerable,
+the greater is the radius of the circle in comparison with the radius <i>R</i> of the
+“world-sphere.” By means of this relation the spherical beings can
+determine the radius of their universe (“world”), even when only a
+relatively small part of their worldsphere is available for their measurements.
+But if this part is very small indeed, they will no longer be able to
+demonstrate that they are on a spherical “world” and not on a
+Euclidean plane, for a small part of a spherical surface differs only slightly
+from a piece of a plane of the same size.
+</p>
+
+<p>
+Thus if the spherical surface beings are living on a planet of which the solar
+system occupies only a negligibly small part of the spherical universe, they
+have no means of determining whether they are living in a finite or in an
+infinite universe, because the “piece of universe” to which they
+have access is in both cases practically plane, or Euclidean. It follows
+directly from this discussion, that for our sphere-beings the circumference of
+a circle first increases with the radius until the “circumference of the
+universe” is reached, and that it thenceforward gradually decreases to
+zero for still further increasing values of the radius. During this process the
+area of the circle continues to increase more and more, until finally it
+becomes equal to the total area of the whole “world-sphere.”
+</p>
+
+<p>
+Perhaps the reader will wonder why we have placed our “beings” on a
+sphere rather than on another closed surface. But this choice has its
+justification in the fact that, of all closed surfaces, the sphere is unique in
+possessing the property that all points on it are equivalent. I admit that the
+ratio of the circumference <i>c</i> of a circle to its radius <i>r</i> depends
+on <i>r</i>, but for a given value of <i>r</i> it is the same for all points of
+the “worldsphere”; in other words, the “world-sphere”
+is a “surface of constant curvature.”
+</p>
+
+<p>
+To this two-dimensional sphere-universe there is a three-dimensional analogy,
+namely, the three-dimensional spherical space which was discovered by Riemann.
+its points are likewise all equivalent. It possesses a finite volume, which is
+determined by its “radius” (2&#960;<sup>2</sup><i>R</i><sup>3</sup>). Is it
+possible to imagine a spherical space? To imagine a space means nothing else
+than that we imagine an epitome of our “space” experience, <i>i.e.</i> of
+experience that we can have in the movement of “rigid” bodies. In
+this sense we <i>can</i> imagine a spherical space.
+</p>
+
+<p>
+Suppose we draw lines or stretch strings in all directions from a point, and
+mark off from each of these the distance <i>r</i> with a measuring-rod. All the free
+end-points of these lengths lie on a spherical surface. We can specially
+measure up the area (<i>F</i>) of this surface by means of a square made up of
+measuring-rods. If the universe is Euclidean, then
+<i>F</i> = 4&#960;<i>r</i><sup>2</sup>; if it is spherical, then <i>F</i> is always less
+than 4&#960;<i>r</i><sup>2</sup>. With increasing values of <i>r, F</i> increases from zero
+up to a maximum value which is determined by the “world-radius,” but
+for still further increasing values of <i>r</i>, the area gradually diminishes to
+zero. At first, the straight lines which radiate from the starting point
+diverge farther and farther from one another, but later they approach each
+other, and finally they run together again at a “counter-point” to
+the starting point. Under such conditions they have traversed the whole
+spherical space. It is easily seen that the three-dimensional spherical space
+is quite analogous to the two-dimensional spherical surface. It is finite (<i>i.e.</i>
+of finite volume), and has no bounds.
+</p>
+
+<p>
+It may be mentioned that there is yet another kind of curved space:
+“elliptical space.” It can be regarded as a curved space in which the
+two “counter-points” are identical (indistinguishable from each
+other). An elliptical universe can thus be considered to some extent as a
+curved universe possessing central symmetry.
+</p>
+
+<p>
+It follows from what has been said, that closed spaces without limits are
+conceivable. From amongst these, the spherical space (and the elliptical)
+excels in its simplicity, since all points on it are equivalent. As a result of
+this discussion, a most interesting question arises for astronomers and
+physicists, and that is whether the universe in which we live is infinite, or
+whether it is finite in the manner of the spherical universe. Our experience is
+far from being sufficient to enable us to answer this question. But the general
+theory of relativity permits of our answering it with a moderate degree of
+certainty, and in this connection the difficulty mentioned in Section XXX finds
+its solution.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap32"></a>XXXII.<br/>
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY</h3>
+
+<p>
+According to the general theory of relativity, the geometrical properties of
+space are not independent, but they are determined by matter. Thus we can draw
+conclusions about the geometrical structure of the universe only if we base our
+considerations on the state of the matter as being something that is known. We
+know from experience that, for a suitably chosen co-ordinate system, the
+velocities of the stars are small as compared with the velocity of transmission
+of light. We can thus as a rough approximation arrive at a conclusion as to the
+nature of the universe as a whole, if we treat the matter as being at rest.
+</p>
+
+<p>
+We already know from our previous discussion that the behaviour of
+measuring-rods and clocks is influenced by gravitational fields, <i>i.e.</i> by the
+distribution of matter. This in itself is sufficient to exclude the possibility
+of the exact validity of Euclidean geometry in our universe. But it is
+conceivable that our universe differs only slightly from a Euclidean one, and
+this notion seems all the more probable, since calculations show that the
+metrics of surrounding space is influenced only to an exceedingly small extent
+by masses even of the magnitude of our sun. We might imagine that, as regards
+geometry, our universe behaves analogously to a surface which is irregularly
+curved in its individual parts, but which nowhere departs appreciably from a
+plane: something like the rippled surface of a lake. Such a universe might
+fittingly be called a quasi-Euclidean universe. As regards its space it would
+be infinite. But calculation shows that in a quasi-Euclidean universe the
+average density of matter would necessarily be <i>nil</i>. Thus such a universe could
+not be inhabited by matter everywhere; it would present to us that
+unsatisfactory picture which we portrayed in Section XXX.
+</p>
+
+<p>
+If we are to have in the universe an average density of matter which differs
+from zero, however small may be that difference, then the universe cannot be
+quasi-Euclidean. On the contrary, the results of calculation indicate that if
+matter be distributed uniformly, the universe would necessarily be spherical
+(or elliptical). Since in reality the detailed distribution of matter is not
+uniform, the real universe will deviate in individual parts from the spherical,
+<i>i.e.</i> the universe will be quasi-spherical. But it will be necessarily finite.
+In fact, the theory supplies us with a simple connection<a href="#linknote-25" name="linknoteref-25" id="linknoteref-25">[25]</a> between
+the space-expanse of the universe and the average density of matter in it.
+</p>
+
+<p>
+<a name="linknote-25" id="linknote-25">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-25"> [25]</a><br/> For the radius <i>R</i> of the
+universe we obtain the equation
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image037.jpg" style="width:100%;" alt="image037" /><br/><br/>
+</div>
+
+<p class="footnote">
+The use of the C.G.S. system in this equation gives 2/k = 1.08 x
+10<sup>27</sup>; &#961; is the average density of the matter and <i>k</i> is a
+constant connected with the Newtonian constant of gravitation.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap33"></a>APPENDICES</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap34"></a>APPENDIX I<br/>
+SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION<br/>
+(SUPPLEMENTARY TO SECTION XI)</h3>
+
+<p>
+For the relative orientation of the co-ordinate systems indicated in Fig. 2,
+the <i>x</i>-axes of both systems permanently coincide. In the present case we can
+divide the problem into parts by considering first only events which are
+localised on the <i>x</i>-axis. Any such event is represented with respect to the
+co-ordinate system <i>K</i> by the abscissa <i>x</i> and the time <i>t</i>, and with respect to the
+system <i>K&#x2032;</i> by the abscissa <i>x&#x2032;</i> and the time <i>t&#x2032;</i>. We require to find <i>x&#x2032;</i> and <i>t&#x2032;</i> when
+<i>x</i> and <i>t</i> are given.
+</p>
+
+<p>
+A light-signal, which is proceeding along the positive axis of <i>x</i>, is
+transmitted according to the equation
+</p>
+
+<p class="center">
+<i>x</i> = <i>ct</i>
+</p>
+
+<p class="noindent">
+or
+</p>
+
+<p class="center">
+<i>x</i> – <i>ct</i> = 0 . . . . . (1).
+</p>
+
+<p class="noindent">
+Since the same light-signal has to be transmitted relative to <i>K&#x2032;</i> with the
+velocity <i>c</i>, the propagation relative to the system <i>K&#x2032;</i> will be represented by
+the analogous formula
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> – <i>ct&#x2032;</i> = 0 . . . . . (2)
+</p>
+
+<p class="noindent">
+Those space-time points (events) which satisfy (1) must also satisfy (2).
+Obviously this will be the case when the relation
+</p>
+
+<p class="center">
+(<i>x&#x2032;</i> – <i>ct&#x2032;</i>) = &#955;(<i>x</i> – <i>ct</i>) . . . (3).
+</p>
+
+<p class="noindent">
+is fulfilled in general, where &#955; indicates a constant; for, according to
+(3), the disappearance of (<i>x</i> – <i>ct</i>) involves the disappearance of (<i>x&#x2032;</i> – <i>ct&#x2032;</i>).
+</p>
+
+<p>
+If we apply quite similar considerations to light rays which are being
+transmitted along the negative <i>x</i>-axis, we obtain the condition
+</p>
+
+<p class="center">
+(<i>x&#x2032;</i> + <i>ct&#x2032;</i>) = (<i>x + ct</i>) . . . (4).
+</p>
+
+<p>
+By adding (or subtracting) equations (3) and (4), and introducing for
+convenience the constants <i>a</i> and <i>b</i> in place of the constants &#955; and &#956; where
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image038.jpg" style="width:100%;" alt="image038" /><br/><br/>
+</div>
+
+<p class="noindent">
+and
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image039.jpg" style="width:100%;" alt="image039" /><br/><br/>
+</div>
+
+<p class="noindent">
+we obtain the equations
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image040.jpg" style="width:100%;" alt="image040" /><br/><br/>
+</div>
+
+<p>
+We should thus have the solution of our problem, if the constants <i>a</i> and <i>b</i> were
+known. These result from the following discussion.
+</p>
+
+<p>
+For the origin of <i>K&#x2032;</i> we have permanently <i>x&#x2032;</i> = 0, and hence according to the
+first of the equations (5)
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image041.jpg" style="width:100%;" alt="image041" /><br/><br/>
+</div>
+
+<p>
+If we call <i>v</i> the velocity with which the origin of <i>K&#x2032;</i> is moving relative to <i>K</i>,
+we then have
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image042.jpg" style="width:100%;" alt="image042" /><br/><br/>
+</div>
+
+<p>
+The same value <i>v</i> can be obtained from equations (5), if we calculate the
+velocity of another point of <i>K&#x2032;</i> relative to <i>K</i>, or the velocity (directed
+towards the negative <i>x</i>-axis) of a point of <i>K</i> with respect to <i>K&#x2032;</i>. In short, we
+can designate <i>v</i> as the relative velocity of the two systems.
+</p>
+
+<p>
+Furthermore, the principle of relativity teaches us that, as judged from K, the
+length of a unit measuring-rod which is at rest with reference to <i>K&#x2032;</i> must be
+exactly the same as the length, as judged from <i>K&#x2032;</i>, of a unit measuring-rod
+which is at rest relative to <i>K</i>. In order to see how the points of the <i>x&#x2032;</i>-axis
+appear as viewed from <i>K</i>, we only require to take a “snapshot” of <i>K&#x2032;</i>
+from <i>K</i>; this means that we have to insert a particular value of <i>t</i> (time of <i>K</i>),
+<i>e.g.</i> <i>t</i> = 0. For this value of <i>t</i> we then obtain from the first of the equations
+(5)
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>ax</i>
+</p>
+
+<p>
+Two points of the <i>x&#x2032;</i>-axis which are separated by the distance &#916;<i>x&#x2032;</i> = 1 when
+measured in the <i>K&#x2032;</i> system are thus separated in our instantaneous photograph by
+the distance
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image043.jpg" style="width:100%;" alt="image043" /><br/><br/>
+</div>
+
+<p>
+But if the snapshot be taken from <i>K&#x2032;</i>(<i>t&#x2032;</i> = 0), and if we eliminate <i>t</i> from the
+equations (5), taking into account the expression (6), we obtain
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image044.jpg" style="width:100%;" alt="image044" /><br/><br/>
+</div>
+
+<p>
+From this we conclude that two points on the <i>x</i>-axis separated by the distance 1
+(relative to <i>K</i>) will be represented on our snapshot by the distance
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image045.jpg" style="width:100%;" alt="image045" /><br/><br/>
+</div>
+
+<p>
+But from what has been said, the two snapshots must be identical; hence &#916;<i>x</i>
+in (7) must be equal to &#916;<i>x&#x2032;</i> in (7<i>a</i>), so that we obtain
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image046.jpg" style="width:100%;" alt="image046" /><br/><br/>
+</div>
+
+<p>
+The equations (6) and (7<i>b</i>) determine the constants <i>a</i> and <i>b</i>. By inserting the
+values of these constants in (5), we obtain the first and the fourth of the
+equations given in Section XI.
+</p>
+
+<div class="fig" style="width:50%;">
+<img src="images/image047.jpg" style="width:100%;" alt="image047" /><br/><br/>
+</div>
+
+<p>
+Thus we have obtained the Lorentz transformation for events on the <i>x</i>-axis. It
+satisfies the condition
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup> = <i>x</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>t</i><sup>2</sup> . . . . . . (8a).
+</p>
+
+<p>
+The extension of this result, to include events which take place outside the
+<i>x</i>-axis, is obtained by retaining equations (8) and supplementing them by the
+relations
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image048.jpg" style="width:100%;" alt="image048" /><br/><br/>
+</div>
+
+<p class="noindent">
+In this way we satisfy the postulate of the constancy of the velocity of light
+<i>in vacuo</i> for rays of light of arbitrary direction, both for the system <i>K</i> and
+for the system <i>K&#x2032;</i>. This may be shown in the following manner.
+</p>
+
+<p>
+We suppose a light-signal sent out from the origin of <i>K</i> at the time <i>t</i> = 0. It
+will be propagated according to the equation
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image049.jpg" style="width:100%;" alt="image049" /><br/><br/>
+</div>
+
+<p class="noindent">
+or, if we square this equation, according to the equation
+</p>
+
+<p class="center">
+<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t</i><sup>2</sup> = 0
+. . . . . (10).
+</p>
+
+<p>
+It is required by the law of propagation of light, in conjunction with the
+postulate of relativity, that the transmission of the signal in question should
+take place—as judged from <i>K&#x2032;</i>—in accordance with the corresponding formula
+</p>
+
+<p class="center">
+<i>r&#x2032;</i> = <i>ct&#x2032;</i>
+</p>
+
+<p class="noindent">
+or,
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= 0 . . . . . . (10<i>a</i>).
+</p>
+
+<p class="noindent">
+In order that equation (10<i>a</i>) may be a consequence of equation (10), we must
+have
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= &#963; (<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>t</i><sup>2</sup>) (11).
+</p>
+
+<p>
+Since equation (8<i>a</i>) must hold for points on the <i>x</i>-axis, we thus have &#963; = 1. It
+is easily seen that the Lorentz transformation really satisfies equation (11)
+for &#963; = 1; for (11) is a consequence of (8<i>a</i>) and (9), and hence also of (8) and
+(9). We have thus derived the Lorentz transformation.
+</p>
+
+<p>
+The Lorentz transformation represented by (8) and (9) still requires to be
+generalised. Obviously it is immaterial whether the axes of <i>K&#x2032;</i> be chosen so
+that they are spatially parallel to those of <i>K</i>. It is also not essential that
+the velocity of translation of <i>K&#x2032;</i> with respect to <i>K</i> should be in the direction
+of the <i>x</i>-axis. A simple consideration shows that we are able to construct the
+Lorentz transformation in this general sense from two kinds of transformations,
+viz. from Lorentz transformations in the special sense and from purely spatial
+transformations. which corresponds to the replacement of the rectangular
+co-ordinate system by a new system with its axes pointing in other directions.
+</p>
+
+<p>
+Mathematically, we can characterise the generalised Lorentz transformation thus:
+</p>
+
+<p>
+It expresses <i>x&#x2032;, y&#x2032;, x&#x2032;, t&#x2032;</i>, in terms of linear homogeneous functions of <i>x, y,
+x, t</i>, of such a kind that the relation
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t</i><sup>2</sup>
+(11<i>a</i>).
+</p>
+
+<p>
+is satisficd identically. That is to say: If we substitute their expressions in
+<i>x, y, x, t</i>, in place of <i>x&#x2032;, y&#x2032;, x&#x2032;, t&#x2032;</i>, on the left-hand side, then the
+left-hand side of (11<i>a</i>) agrees with the right-hand side.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap35"></a>APPENDIX II<br/>
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)<br/>
+(SUPPLEMENTARY TO SECTION XVII)</h3>
+
+<p>
+We can characterise the Lorentz transformation still more simply if we
+introduce the imaginary
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image031.jpg" style="width:100%;" alt="image031" /><br/><br/>
+</div>
+
+<p class="noindent">
+in place of <i>t</i>, as time-variable. If, in accordance with this, we insert
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image050.jpg" style="width:100%;" alt="image050" /><br/><br/>
+</div>
+
+<p class="noindent">
+and similarly for the accented system <i>K&#x2032;</i>, then the condition which is
+identically satisfied by the transformation can be expressed thus:
+</p>
+
+<p class="center">
+<i>x</i><sub>1</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>2</sub>&#x2032;<sup>2</sup> +
+<i>x</i><sub>3</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>4</sub>&#x2032;<sup>2</sup> =
+<i>x</i><sub>1</sub><sup>2</sup> + <i>x</i><sub>2</sub><sup>2</sup> +
+<i>x</i><sub>3</sub><sup>2</sup> + <i>x</i><sub>4</sub><sup>2 </sup>(12).
+</p>
+
+<p>
+That is, by the afore-mentioned choice of “coordinates,” (11<i>a</i>) [see
+the end of Appendix II] is transformed into this equation.
+</p>
+
+<p>
+We see from (12) that the imaginary time co-ordinate <i>x</i><sub>4</sub>, enters into
+the condition of transformation in exactly the same way as the space
+co-ordinates <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>. It is due to this
+fact that, according to the theory of relativity, the “time”
+<i>x</i><sub>4</sub>, enters into natural laws in the same form as the space co
+ordinates <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>.
+</p>
+
+<p>
+A four-dimensional continuum described by the “co-ordinates”
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, was called
+“world” by Minkowski, who also termed a point-event a
+“world-point.” From a “happening” in three-dimensional
+space, physics becomes, as it were, an “existence” in the
+four-dimensional “world.”
+</p>
+
+<p>
+This four-dimensional “world” bears a close similarity to the
+three-dimensional “space” of (Euclidean) analytical geometry. If we
+introduce into the latter a new Cartesian co-ordinate system (<i>x&#x2032;</i><sub>1</sub>,
+<i>x&#x2032;</i><sub>2</sub>, <i>x&#x2032;</i><sub>3</sub>) with the same origin, then <i>x&#x2032;</i><sub>1</sub>,
+<i>x&#x2032;</i><sub>2</sub>, <i>x&#x2032;</i><sub>3</sub>, are linear homogeneous functions of
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub> which identically satisfy the
+equation
+</p>
+
+<p class="center">
+<i>x</i><sub>1</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>2</sub>&#x2032;<sup>2</sup> +
+<i>x</i><sub>3</sub>&#x2032;<sup>2</sup> = <i>x</i><sub>1</sub><sup>2</sup> +
+<i>x</i><sub>2</sub><sup>2</sup> + <i>x</i><sub>3</sub><sup>2</sup>
+</p>
+
+<p class="noindent">
+The analogy with (12) is a complete one. We can regard Minkowski’s
+“world” in a formal manner as a four-dimensional Euclidean space
+(with an imaginary time coordinate); the Lorentz transformation corresponds
+to a “rotation” of the co-ordinate system in the four-dimensional
+“world.”
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap36"></a>APPENDIX III<br/>
+THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY</h3>
+
+<p>
+From a systematic theoretical point of view, we may imagine the process of
+evolution of an empirical science to be a continuous process of induction.
+Theories are evolved and are expressed in short compass as statements of a
+large number of individual observations in the form of empirical laws, from
+which the general laws can be ascertained by comparison. Regarded in this way,
+the development of a science bears some resemblance to the compilation of a
+classified catalogue. It is, as it were, a purely empirical enterprise.
+</p>
+
+<p>
+But this point of view by no means embraces the whole of the actual process;
+for it slurs over the important part played by intuition and deductive thought
+in the development of an exact science. As soon as a science has emerged from
+its initial stages, theoretical advances are no longer achieved merely by a
+process of arrangement. Guided by empirical data, the investigator rather
+develops a system of thought which, in general, is built up logically from a
+small number of fundamental assumptions, the so-called axioms. We call such a
+system of thought a <i>theory</i>. The theory finds the justification for its
+existence in the fact that it correlates a large number of single observations,
+and it is just here that the “truth” of the theory lies.
+</p>
+
+<p>
+Corresponding to the same complex of empirical data, there may be several
+theories, which differ from one another to a considerable extent. But as
+regards the deductions from the theories which are capable of being tested, the
+agreement between the theories may be so complete that it becomes difficult to
+find any deductions in which the two theories differ from each other. As an
+example, a case of general interest is available in the province of biology, in
+the Darwinian theory of the development of species by selection in the struggle
+for existence, and in the theory of development which is based on the
+hypothesis of the hereditary transmission of acquired characters.
+</p>
+
+<p>
+We have another instance of far-reaching agreement between the deductions from
+two theories in Newtonian mechanics on the one hand, and the general theory of
+relativity on the other. This agreement goes so far, that up to the present we
+have been able to find only a few deductions from the general theory of
+relativity which are capable of investigation, and to which the physics of
+pre-relativity days does not also lead, and this despite the profound
+difference in the fundamental assumptions of the two theories. In what follows,
+we shall again consider these important deductions, and we shall also discuss
+the empirical evidence appertaining to them which has hitherto been obtained.
+</p>
+
+<h4>
+(<i>a</i>) Motion of the Perihelion of Mercury
+</h4>
+
+<p>
+According to Newtonian mechanics and Newton’s law of gravitation, a planet
+which is revolving round the sun would describe an ellipse round the latter,
+or, more correctly, round the common centre of gravity of the sun and the
+planet. In such a system, the sun, or the common centre of gravity, lies in one
+of the foci of the orbital ellipse in such a manner that, in the course of a
+planet-year, the distance sun-planet grows from a minimum to a maximum, and
+then decreases again to a minimum. If instead of Newton’s law we insert a
+somewhat different law of attraction into the calculation, we find that,
+according to this new law, the motion would still take place in such a manner
+that the distance sun-planet exhibits periodic variations; but in this case the
+angle described by the line joining sun and planet during such a period (from
+perihelion—closest proximity to the sun—to perihelion) would differ from 360&deg;.
+The line of the orbit would not then be a closed one but in the course of time
+it would fill up an annular part of the orbital plane, viz. between the circle
+of least and the circle of greatest distance of the planet from the sun.
+</p>
+
+<p>
+According also to the general theory of relativity, which differs of course
+from the theory of Newton, a small variation from the Newton-Kepler motion of a
+planet in its orbit should take place, and in such away, that the angle
+described by the radius sun-planet between one perhelion and the next should
+exceed that corresponding to one complete revolution by an amount given by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image051.jpg" style="width:100%;" alt="image051" /><br/><br/>
+</div>
+
+<p>
+(<i>N.B.</i>—One complete revolution corresponds to the angle 2&#960; in the absolute
+angular measure customary in physics, and the above expression given the amount
+by which the radius sun-planet exceeds this angle during the interval between
+one perihelion and the next.) In this expression <i>a</i> represents the major
+semi-axis of the ellipse, <i>e</i> its eccentricity, <i>c</i> the velocity of light, and <i>T</i>
+the period of revolution of the planet. Our result may also be stated as
+follows: According to the general theory of relativity, the major axis of the
+ellipse rotates round the sun in the same sense as the orbital motion of the
+planet. Theory requires that this rotation should amount to 43 seconds of arc
+per century for the planet Mercury, but for the other Planets of our solar
+system its magnitude should be so small that it would necessarily escape
+detection.<a href="#linknote-26" name="linknoteref-26" id="linknoteref-26">[26]</a>
+</p>
+
+<p>
+<a name="linknote-26" id="linknote-26">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-26"> [26]</a><br/> Especially since the next planet Venus
+has an orbit that is almost an exact circle, which makes it more difficult to
+locate the perihelion with precision.
+</p>
+
+<p>
+In point of fact, astronomers have found that the theory of Newton does not
+suffice to calculate the observed motion of Mercury with an exactness
+corresponding to that of the delicacy of observation attainable at the present
+time. After taking account of all the disturbing influences exerted on Mercury
+by the remaining planets, it was found (Leverrier: 1859; and Newcomb: 1895)
+that an unexplained perihelial movement of the orbit of Mercury remained over,
+the amount of which does not differ sensibly from the above mentioned +43
+seconds of arc per century. The uncertainty of the empirical result amounts to
+a few seconds only.
+</p>
+
+<h4>
+(<i>b</i>) Deflection of Light by a Gravitational Field
+</h4>
+
+<div class="fig" style="width:20%;">
+<img src="images/image052.jpg" style="width:100%;" alt="image052" /><br/><br/>
+</div>
+
+<p>
+In Section XXII it has been already mentioned that according to the general
+theory of relativity, a ray of light will experience a curvature of its path
+when passing through a gravitational field, this curvature being similar to
+that experienced by the path of a body which is projected through a
+gravitational field. As a result of this theory, we should expect that a ray of
+light which is passing close to a heavenly body would be deviated towards the
+latter. For a ray of light which passes the sun at a distance of &#916;
+sun-radii from its centre, the angle of deflection (&#945;) should amount to
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image053.jpg" style="width:100%;" alt="image053" /><br/><br/>
+</div>
+
+<p>
+It may be added that, according to the theory, half of this deflection is
+produced by the Newtonian field of attraction of the sun, and the other half by
+the geometrical modification (“curvature”) of space caused by the
+sun.
+</p>
+
+<p>
+This result admits of an experimental test by means of the photographic
+registration of stars during a total eclipse of the sun. The only reason why we
+must wait for a total eclipse is because at every other time the atmosphere is
+so strongly illuminated by the light from the sun that the stars situated near
+the sun’s disc are invisible. The predicted effect can be seen clearly from the
+accompanying diagram. If the sun (<i>S</i>) were not present, a star which is
+practically infinitely distant would be seen in the direction <i>D</i><sub>1</sub>, as
+observed front the earth. But as a consequence of the deflection of light from
+the star by the sun, the star will be seen in the direction <i>D</i><sub>2</sub>, <i>i.e.</i>
+at a somewhat greater distance from the centre of the sun than corresponds to
+its real position.
+</p>
+
+<p>
+In practice, the question is tested in the following way. The stars in the
+neighbourhood of the sun are photographed during a solar eclipse.
+</p>
+
+<p>
+In addition, a second photograph of the same stars is taken when the sun is
+situated at another position in the sky, <i>i.e.</i> a few months earlier or
+later. As compared with the standard photograph, the positions of the stars on
+the eclipse-photograph ought to appear displaced radially outwards (away from
+the centre of the sun) by an amount corresponding to the angle <i>a</i>.
+</p>
+
+<p>
+We are indebted to the [British] Royal Society and to the Royal Astronomical
+Society for the investigation of this important deduction. Undaunted by the
+[first world] war and by difficulties of both a material and a psychological
+nature aroused by the war, these societies equipped two expeditions—to Sobral
+(Brazil), and to the island of Principe (West Africa)—and sent several of
+Britain’s most celebrated astronomers (Eddington, Cottingham, Crommelin,
+Davidson), in order to obtain photographs of the solar eclipse of 29th May,
+1919. The relative discrepancies to be expected between the stellar photographs
+obtained during the eclipse and the comparison photographs amounted to a few
+hundredths of a millimetre only. Thus great accuracy was necessary in making
+the adjustments required for the taking of the photographs, and in their
+subsequent measurement.
+</p>
+
+<p>
+The results of the measurements confirmed the theory in a thoroughly
+satisfactory manner. The rectangular components of the observed and of the
+calculated deviations of the stars (in seconds of arc) are set forth in the
+following table of results:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image054.jpg" style="width:100%;" alt="image054" /><br/><br/>
+</div>
+
+<h4>
+(<i>c</i>) Displacement of Spectral Lines Towards the Red
+</h4>
+
+<p>
+In Section XXIII it has been shown that in a system <i>K&#x2032;</i> which is in rotation with
+regard to a Galileian system <i>K</i>, clocks of identical construction, and which are
+considered at rest with respect to the rotating reference-body, go at rates
+which are dependent on the positions of the clocks. We shall now examine this
+dependence quantitatively. A clock, which is situated at a distance r from the
+centre of the disc, has a velocity relative to <i>K</i> which is given by
+</p>
+
+<p class="center">
+<i>v</i> = &#969;<i>r</i>,
+</p>
+
+<p class="noindent">
+where &#969; represents the angular velocity of rotation of the disc <i>K&#x2032;</i> with respect
+to <i>K</i>. If <i>v</i><sub>0</sub>, represents the number of ticks of the clock per unit
+time (“rate” of the clock) relative to <i>K</i> when the clock is at rest,
+then the “rate” of the clock (<i>v</i>) when it is moving relative to <i>K</i> with
+a velocity <i>v</i>, but at rest with respect to the disc, will, in accordance with
+Section XII, be given by
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image055.jpg" style="width:100%;" alt="image055" /><br/><br/>
+</div>
+
+<p class="noindent">
+or with sufficient accuracy by
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image056.jpg" style="width:100%;" alt="image056" /><br/><br/>
+</div>
+
+<p class="noindent">
+This expression may also be stated in the following form:
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image057.jpg" style="width:100%;" alt="image057" /><br/><br/>
+</div>
+
+<p class="noindent">
+If we represent the difference of potential of the centrifugal force between
+the position of the clock and the centre of the disc by &#966;, <i>i.e.</i> the work,
+considered negatively, which must be performed on the unit of mass against the
+centrifugal force in order to transport it from the position of the clock on
+the rotating disc to the centre of the disc, then we have
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image058.jpg" style="width:100%;" alt="image058" /><br/><br/>
+</div>
+
+<p class="noindent">
+From this it follows that
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image059.jpg" style="width:100%;" alt="image059" /><br/><br/>
+</div>
+
+<p class="noindent">
+In the first place, we see from this expression that two clocks of identical
+construction will go at different rates when situated at different distances
+from the centre of the disc. This result is also valid from the standpoint of
+an observer who is rotating with the disc.
+</p>
+
+<p>
+Now, as judged from the disc, the latter is in a gravitational field of
+potential &#966;, hence the result we have obtained will hold quite generally for
+gravitational fields. Furthermore, we can regard an atom which is emitting
+spectral lines as a clock, so that the following statement will hold:
+</p>
+
+<p>
+<i>An atom absorbs or emits light of a frequency which is dependent on the
+potential of the gravitational field in which it is situated.</i>
+</p>
+
+<p>
+The frequency of an atom situated on the surface of a heavenly body will be
+somewhat less than the frequency of an atom of the same element which is
+situated in free space (or on the surface of a smaller celestial body).
+</p>
+
+<p>
+Now &#966; = – <i>K (M/r)</i>, where <i>K</i> is Newton’s constant of gravitation, and <i>M</i> is the
+mass of the heavenly body. Thus a displacement towards the red ought to take
+place for spectral lines produced at the surface of stars as compared with the
+spectral lines of the same element produced at the surface of the earth, the
+amount of this displacement being
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image060.jpg" style="width:100%;" alt="image060" /><br/><br/>
+</div>
+
+<p>
+For the sun, the displacement towards the red predicted by theory amounts to
+about two millionths of the wave-length. A trustworthy calculation is not
+possible in the case of the stars, because in general neither the mass <i>M</i> nor
+the radius <i>r</i> are known.
+</p>
+
+<p>
+It is an open question whether or not this effect exists, and at the present
+time (1920) astronomers are working with great zeal towards the solution. Owing
+to the smallness of the effect in the case of the sun, it is difficult to form
+an opinion as to its existence. Whereas Grebe and Bachem (Bonn), as a result of
+their own measurements and those of Evershed and Schwarzschild on the cyanogen
+bands, have placed the existence of the effect almost beyond doubt, while other
+investigators, particularly St. John, have been led to the opposite opinion in
+consequence of their measurements.
+</p>
+
+<p>
+Mean displacements of lines towards the less refrangible end of the spectrum
+are certainly revealed by statistical investigations of the fixed stars; but
+up to the present the examination of the available data does not allow of any
+definite decision being arrived at, as to whether or not these displacements
+are to be referred in reality to the effect of gravitation. The results of
+observation have been collected together, and discussed in detail from the
+standpoint of the question which has been engaging our attention here, in a
+paper by E. Freundlich entitled “Zur Prüfung der allgemeinen
+Relativitäts-Theorie” (<i>Die Naturwissenschaften</i>, 1919, No. 35, p. 520:
+Julius Springer, Berlin).
+</p>
+
+<p>
+At all events, a definite decision will be reached during the next few years.
+If the displacement of spectral lines towards the red by the gravitational
+potential does not exist, then the general theory of relativity will be
+untenable. On the other hand, if the cause of the displacement of spectral
+lines be definitely traced to the gravitational potential, then the study of
+this displacement will furnish us with important information as to the mass of
+the heavenly bodies.<a href="#linknote-27" name="linknoteref-27" id="linknoteref-27">[27]</a>
+</p>
+
+<p>
+<a name="linknote-27" id="linknote-27">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-27"> [27]</a><br/> The displacement of spectral lines
+towards the red end of the spectrum was definitely established by Adams in
+1924, by observations on the dense companion of Sirius, for which the effect is
+about thirty times greater than for the Sun. R.W.L.—translator
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap37"></a>APPENDIX IV<br/>
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY<br/>
+(SUPPLEMENTARY TO SECTION XXXII)
+</h3>
+
+<p>
+Since the publication of the first edition of this little book, our knowledge
+about the structure of space in the large (“cosmological problem”)
+has had an important development, which ought to be mentioned even in a popular
+presentation of the subject.
+</p>
+
+<p>
+My original considerations on the subject were based on two hypotheses:
+</p>
+
+<p>
+(1) There exists an average density of matter in the whole of space which is
+everywhere the same and different from zero.
+</p>
+
+<p>
+(2) The magnitude (“radius”) of space is independent of time.
+</p>
+
+<p>
+Both these hypotheses proved to be consistent, according to the general theory
+of relativity, but only after a hypothetical term was added to the field
+equations, a term which was not required by the theory as such nor did it seem
+natural from a theoretical point of view (“cosmological term of the field
+equations”).
+</p>
+
+<p>
+Hypothesis (2) appeared unavoidable to me at the time, since I thought that one
+would get into bottomless speculations if one departed from it.
+</p>
+
+<p>
+However, already in the ’twenties, the Russian mathematician Friedman showed
+that a different hypothesis was natural from a purely theoretical point of
+view. He realized that it was possible to preserve hypothesis (1) without
+introducing the less natural cosmological term into the field equations of
+gravitation, if one was ready to drop hypothesis (2). Namely, the original
+field equations admit a solution in which the “world radius” depends
+on time (expanding space). In that sense one can say, according to Friedman,
+that the theory demands an expansion of space.
+</p>
+
+<p>
+A few years later Hubble showed, by a special investigation of the
+extra-galactic nebulae (“milky ways”), that the spectral lines
+emitted showed a red shift which increased regularly with the distance of the
+nebulae. This can be interpreted in regard to our present knowledge only in the
+sense of Doppler’s principle, as an expansive motion of the system of stars in
+the large—as required, according to Friedman, by the field equations of
+gravitation. Hubble’s discovery can, therefore, be considered to some extent as
+a confirmation of the theory.
+</p>
+
+<p>
+There does arise, however, a strange difficulty. The interpretation of the
+galactic line-shift discovered by Hubble as an expansion (which can hardly be
+doubted from a theoretical point of view), leads to an origin of this expansion
+which lies “only” about 10<sup>9</sup> years ago, while physical
+astronomy makes it appear likely that the development of individual stars and
+systems of stars takes considerably longer. It is in no way known how this
+incongruity is to be overcome.
+</p>
+
+<p>
+I further want to remark that the theory of expanding space, together with the
+empirical data of astronomy, permit no decision to be reached about the finite
+or infinite character of (three-dimensional) space, while the original
+“static” hypothesis of space yielded the closure (finiteness) of
+space.
+</p>
+
+<p>
+<i>K</i> = co-ordinate system
+</p>
+
+<p>
+<i>x, y</i> = two-dimensional co-ordinates
+</p>
+
+<p>
+<i>x, y, z</i> = three-dimensional co-ordinates
+</p>
+
+<p>
+<i>x, y, z, t</i> = four-dimensional co-ordinates
+</p>
+
+<p>
+<i>t</i> = time
+</p>
+
+<p>
+<i>I</i> = distance
+</p>
+
+<p>
+<i>v</i> = velocity
+</p>
+
+<p>
+<i>F</i> = force
+</p>
+
+<p>
+<i>G</i> = gravitational field
+</p>
+
+</div><!--end chapter-->
+
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+Project Gutenberg (https://www.gutenberg.org) public repository for
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+The Project Gutenberg eBook of Relativity: The Special and General Theory, by Albert Einstein
+
+This eBook is for the use of anyone anywhere in the United States and
+most other parts of the world at no cost and with almost no restrictions
+whatsoever. You may copy it, give it away or re-use it under the terms
+of the Project Gutenberg License included with this eBook or online at
+www.gutenberg.org. If you are not located in the United States, you
+will have to check the laws of the country where you are located before
+using this eBook.
+
+Title: Relativity: The Special and General Theory
+
+Author: Albert Einstein
+
+Release Date: October 1, 2009 [eBook #30155]
+[Most recently updated: May 2, 2023]
+
+Language: English
+
+Produced by: Robert Hux
+
+*** START OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***
+
+
+
+
+cover 
+
+
+
+
+Relativity: The Special and General Theory
+
+by Albert Einstein
+
+
+Authorised Translation by Robert W. Lawson
+
+
+
+
+ALBERT EINSTEIN REFERENCE ARCHIVE
+RELATIVITY: THE SPECIAL AND GENERAL THEORY
+BY ALBERT EINSTEIN
+
+
+Written: 1916 (this revised edition: 1924)
+Source: Relativity: The Special and General Theory (1920)
+Publisher: Methuen & Co Ltd
+First Published: December, 1916
+Translated: Robert W. Lawson (Authorised translation)
+Transcription/Markup: Brian Basgen
+Transcription to text: Gregory B. Newby
+Thanks to: Einstein Reference Archive (marxists.org)
+The Einstein Reference Archive is online at:
+http://www.marxists.org/reference/archive/einstein/index.htm
+
+
+
+
+Contents
+
+ Preface
+
+ Part I: The Special Theory of Relativity
+ I. Physical Meaning of Geometrical Propositions
+ II. The System of Co-ordinates
+ III. Space and Time in Classical Mechanics
+ IV. The Galileian System of Co-ordinates
+ V. The Principle of Relativity (in the Restricted Sense)
+ VI. The Theorem of the Addition of Velocities employed in Classical Mechanics
+ VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity
+ VIII. On the Idea of Time in Physics
+ IX. The Relativity of Simultaneity
+ X. On the Relativity of the Conception of Distance
+ XI. The Lorentz Transformation
+ XII. The Behaviour of Measuring-Rods and Clocks in Motion
+ XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau
+ XIV. The Heuristic Value of the Theory of Relativity
+ XV. General Results of the Theory
+ XVI. Experience and the Special Theory of Relativity
+ XVII. Minkowski’s Four-dimensional Space
+
+ Part II: The General Theory of Relativity
+ XVIII. Special and General Principle of Relativity
+ XIX. The Gravitational Field
+ XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
+ XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
+ XXII. A Few Inferences from the General Principle of Relativity
+ XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
+ XXIV. Euclidean and non-Euclidean Continuum
+ XXV. Gaussian Co-ordinates
+ XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
+ XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum
+ XXVIII. Exact Formulation of the General Principle of Relativity
+ XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
+
+ Part III: Considerations on the Universe as a Whole
+ XXX. Cosmological Difficulties of Newton’s Theory
+ XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe
+ XXXII. The Structure of Space According to the General Theory of Relativity
+
+ Appendices:
+ I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)
+ II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII)
+ III. The Experimental Confirmation of the General Theory of Relativity
+ IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII)
+ V. Relativity and the Problem of Space
+
+
+Note: The fifth Appendix was added by Einstein at the time of the
+fifteenth re-printing of this book; and as a result is still under
+copyright restrictions so cannot be added without the permission of the
+publisher.
+
+
+
+
+PREFACE
+
+
+The present book is intended, as far as possible, to give an exact
+insight into the theory of Relativity to those readers who, from a
+general scientific and philosophical point of view, are interested in
+the theory, but who are not conversant with the mathematical apparatus
+of theoretical physics. The work presumes a standard of education
+corresponding to that of a university matriculation examination, and,
+despite the shortness of the book, a fair amount of patience and force
+of will on the part of the reader. The author has spared himself no
+pains in his endeavour to present the main ideas in the simplest and
+most intelligible form, and on the whole, in the sequence and
+connection in which they actually originated. In the interest of
+clearness, it appeared to me inevitable that I should repeat myself
+frequently, without paying the slightest attention to the elegance of
+the presentation. I adhered scrupulously to the precept of that
+brilliant theoretical physicist L. Boltzmann, according to whom matters
+of elegance ought to be left to the tailor and to the cobbler. I make
+no pretence of having withheld from the reader difficulties which are
+inherent to the subject. On the other hand, I have purposely treated
+the empirical physical foundations of the theory in a “step-motherly”
+fashion, so that readers unfamiliar with physics may not feel like the
+wanderer who was unable to see the forest for the trees. May the book
+bring some one a few happy hours of suggestive thought!
+
+December, 1916
+
+ A. EINSTEIN
+
+
+
+
+PART I: THE SPECIAL THEORY OF RELATIVITY
+
+
+
+
+I.
+PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
+
+
+In your schooldays most of you who read this book made acquaintance
+with the noble building of Euclid’s geometry, and you remember—perhaps
+with more respect than love—the magnificent structure, on the lofty
+staircase of which you were chased about for uncounted hours by
+conscientious teachers. By reason of our past experience, you would
+certainly regard everyone with disdain who should pronounce even the
+most out-of-the-way proposition of this science to be untrue. But
+perhaps this feeling of proud certainty would leave you immediately if
+some one were to ask you: “What, then, do you mean by the assertion
+that these propositions are true?” Let us proceed to give this question
+a little consideration.
+
+Geometry sets out from certain conceptions such as “plane,” “point,”
+and “straight line,” with which we are able to associate more or less
+definite ideas, and from certain simple propositions (axioms) which, in
+virtue of these ideas, we are inclined to accept as “true.” Then, on
+the basis of a logical process, the justification of which we feel
+ourselves compelled to admit, all remaining propositions are shown to
+follow from those axioms, _i.e._ they are proven. A proposition is then
+correct (“true”) when it has been derived in the recognised manner from
+the axioms. The question of “truth” of the individual geometrical
+propositions is thus reduced to one of the “truth” of the axioms. Now
+it has long been known that the last question is not only unanswerable
+by the methods of geometry, but that it is in itself entirely without
+meaning. We cannot ask whether it is true that only one straight line
+goes through two points. We can only say that Euclidean geometry deals
+with things called “straight lines,” to each of which is ascribed the
+property of being uniquely determined by two points situated on it. The
+concept “true” does not tally with the assertions of pure geometry,
+because by the word “true” we are eventually in the habit of
+designating always the correspondence with a “real” object; geometry,
+however, is not concerned with the relation of the ideas involved in it
+to objects of experience, but only with the logical connection of these
+ideas among themselves.
+
+It is not difficult to understand why, in spite of this, we feel
+constrained to call the propositions of geometry “true.” Geometrical
+ideas correspond to more or less exact objects in nature, and these
+last are undoubtedly the exclusive cause of the genesis of those ideas.
+Geometry ought to refrain from such a course, in order to give to its
+structure the largest possible logical unity. The practice, for
+example, of seeing in a “distance” two marked positions on a
+practically rigid body is something which is lodged deeply in our habit
+of thought. We are accustomed further to regard three points as being
+situated on a straight line, if their apparent positions can be made to
+coincide for observation with one eye, under suitable choice of our
+place of observation.
+
+If, in pursuance of our habit of thought, we now supplement the
+propositions of Euclidean geometry by the single proposition that two
+points on a practically rigid body always correspond to the same
+distance (line-interval), independently of any changes in position to
+which we may subject the body, the propositions of Euclidean geometry
+then resolve themselves into propositions on the possible relative
+position of practically rigid bodies.[1] Geometry which has been
+supplemented in this way is then to be treated as a branch of physics.
+We can now legitimately ask as to the “truth” of geometrical
+propositions interpreted in this way, since we are justified in asking
+whether these propositions are satisfied for those real things we have
+associated with the geometrical ideas. In less exact terms we can
+express this by saying that by the “truth” of a geometrical proposition
+in this sense we understand its validity for a construction with rule
+and compasses.
+
+
+ [1] It follows that a natural object is associated also with a
+ straight line. Three points _A, B_ and _C_ on a rigid body thus lie in
+ a straight line when the points _A_ and _C_ being given, _B_ is chosen
+ such that the sum of the distances _AB_ and _BC_ is as short as
+ possible. This incomplete suggestion will suffice for the present
+ purpose.
+
+
+Of course the conviction of the “truth” of geometrical propositions in
+this sense is founded exclusively on rather incomplete experience. For
+the present we shall assume the “truth” of the geometrical
+propositions, then at a later stage (in the general theory of
+relativity) we shall see that this “truth” is limited, and we shall
+consider the extent of its limitation.
+
+
+
+
+II.
+THE SYSTEM OF CO-ORDINATES
+
+
+On the basis of the physical interpretation of distance which has been
+indicated, we are also in a position to establish the distance between
+two points on a rigid body by means of measurements. For this purpose
+we require a “distance” (rod _S_) which is to be used once and for all,
+and which we employ as a standard measure. If, now, _A_ and _B_ are two
+points on a rigid body, we can construct the line joining them
+according to the rules of geometry; then, starting from _A_, we can
+mark off the distance _S_ time after time until we reach _B_. The
+number of these operations required is the numerical measure of the
+distance _AB_. This is the basis of all measurement of length.[2]
+
+
+ [2] Here we have assumed that there is nothing left over _i.e._ that
+ the measurement gives a whole number. This difficulty is got over by
+ the use of divided measuring-rods, the introduction of which does not
+ demand any fundamentally new method.
+
+
+Every description of the scene of an event or of the position of an
+object in space is based on the specification of the point on a rigid
+body (body of reference) with which that event or object coincides.
+This applies not only to scientific description, but also to everyday
+life. If I analyse the place specification “Trafalgar Square,
+London”[3] I arrive at the following result. The earth is the rigid
+body to which the specification of place refers; “Trafalgar Square,
+London” is a well-defined point, to which a name has been assigned, and
+with which the event coincides in space.[4]
+
+
+ [3]
+
+I have chosen this as being more familiar to the English reader than
+the “Potzdammer Platz, Berlin,” which is referred to in the original.
+(R. W. L.)
+
+
+ [4] It is not necessary here to investigate further the significance
+ of the expression “coincidence in space.” This conception is
+ sufficiently obvious to ensure that differences of opinion are
+ scarcely likely to arise as to its applicability in practice.
+
+
+This primitive method of place specification deals only with places on
+the surface of rigid bodies, and is dependent on the existence of
+points on this surface which are distinguishable from each other. But
+we can free ourselves from both of these limitations without altering
+the nature of our specification of position. If, for instance, a cloud
+is hovering over Trafalgar Square, then we can determine its position
+relative to the surface of the earth by erecting a pole perpendicularly
+on the Square, so that it reaches the cloud. The length of the pole
+measured with the standard measuring-rod, combined with the
+specification of the position of the foot of the pole, supplies us with
+a complete place specification. On the basis of this illustration, we
+are able to see the manner in which a refinement of the conception of
+position has been developed.
+
+(_a_) We imagine the rigid body, to which the place specification is
+referred, supplemented in such a manner that the object whose position
+we require is reached by the completed rigid body.
+
+(_b_) In locating the position of the object, we make use of a number
+(here the length of the pole measured with the measuring-rod) instead
+of designated points of reference.
+
+(_c_) We speak of the height of the cloud even when the pole which
+reaches the cloud has not been erected. By means of optical
+observations of the cloud from different positions on the ground, and
+taking into account the properties of the propagation of light, we
+determine the length of the pole we should have required in order to
+reach the cloud.
+
+From this consideration we see that it will be advantageous if, in the
+description of position, it should be possible by means of numerical
+measures to make ourselves independent of the existence of marked
+positions (possessing names) on the rigid body of reference. In the
+physics of measurement this is attained by the application of the
+Cartesian system of co-ordinates.
+
+This consists of three plane surfaces perpendicular to each other and
+rigidly attached to a rigid body. Referred to a system of co-ordinates,
+the scene of any event will be determined (for the main part) by the
+specification of the lengths of the three perpendiculars or
+co-ordinates (_x, y, z_) which can be dropped from the scene of the
+event to those three plane surfaces. The lengths of these three
+perpendiculars can be determined by a series of manipulations with
+rigid measuring-rods performed according to the rules and methods laid
+down by Euclidean geometry.
+
+In practice, the rigid surfaces which constitute the system of
+co-ordinates are generally not available; furthermore, the magnitudes
+of the co-ordinates are not actually determined by constructions with
+rigid rods, but by indirect means. If the results of physics and
+astronomy are to maintain their clearness, the physical meaning of
+specifications of position must always be sought in accordance with the
+above considerations.[5]
+
+
+ [5] A refinement and modification of these views does not become
+ necessary until we come to deal with the general theory of relativity,
+ treated in the second part of this book.
+
+
+We thus obtain the following result: Every description of events in
+space involves the use of a rigid body to which such events have to be
+referred. The resulting relationship takes for granted that the laws of
+Euclidean geometry hold for “distances;” the “distance” being
+represented physically by means of the convention of two marks on a
+rigid body.
+
+
+III.
+
+SPACE AND TIME IN CLASSICAL MECHANICS
+
+The purpose of mechanics is to describe how bodies change their
+position in space with “time.” I should load my conscience with grave
+sins against the sacred spirit of lucidity were I to formulate the aims
+of mechanics in this way, without serious reflection and detailed
+explanations. Let us proceed to disclose these sins.
+
+It is not clear what is to be understood here by “position” and
+“space.” I stand at the window of a railway carriage which is
+travelling uniformly, and drop a stone on the embankment, without
+throwing it. Then, disregarding the influence of the air resistance, I
+see the stone descend in a straight line. A pedestrian who observes the
+misdeed from the footpath notices that the stone falls to earth in a
+parabolic curve. I now ask: Do the “positions” traversed by the stone
+lie “in reality” on a straight line or on a parabola? Moreover, what is
+meant here by motion “in space”? From the considerations of the
+previous section the answer is self-evident. In the first place we
+entirely shun the vague word “space,” of which, we must honestly
+acknowledge, we cannot form the slightest conception, and we replace it
+by “motion relative to a practically rigid body of reference.” The
+positions relative to the body of reference (railway carriage or
+embankment) have already been defined in detail in the preceding
+section. If instead of “body of reference” we insert “system of
+co-ordinates,” which is a useful idea for mathematical description, we
+are in a position to say: The stone traverses a straight line relative
+to a system of co-ordinates rigidly attached to the carriage, but
+relative to a system of co-ordinates rigidly attached to the ground
+(embankment) it describes a parabola. With the aid of this example it
+is clearly seen that there is no such thing as an independently
+existing trajectory (lit. “path-curve”[6], but only a trajectory
+relative to a particular body of reference.
+
+
+ [6] That is, a curve along which the body moves.
+
+
+In order to have a _complete_ description of the motion, we must
+specify how the body alters its position _with time; i.e._ for every
+point on the trajectory it must be stated at what time the body is
+situated there. These data must be supplemented by such a definition of
+time that, in virtue of this definition, these time-values can be
+regarded essentially as magnitudes (results of measurements) capable of
+observation. If we take our stand on the ground of classical mechanics,
+we can satisfy this requirement for our illustration in the following
+manner. We imagine two clocks of identical construction; the man at the
+railway-carriage window is holding one of them, and the man on the
+footpath the other. Each of the observers determines the position on
+his own reference-body occupied by the stone at each tick of the clock
+he is holding in his hand. In this connection we have not taken account
+of the inaccuracy involved by the finiteness of the velocity of
+propagation of light. With this and with a second difficulty prevailing
+here we shall have to deal in detail later.
+
+
+IV. THE GALILEIAN SYSTEM OF CO-ORDINATES
+
+
+As is well known, the fundamental law of the mechanics of
+Galilei-Newton, which is known as the _law of inertia_, can be stated
+thus: A body removed sufficiently far from other bodies continues in a
+state of rest or of uniform motion in a straight line. This law not
+only says something about the motion of the bodies, but it also
+indicates the reference-bodies or systems of coordinates, permissible
+in mechanics, which can be used in mechanical description. The visible
+fixed stars are bodies for which the law of inertia certainly holds to
+a high degree of approximation. Now if we use a system of co-ordinates
+which is rigidly attached to the earth, then, relative to this system,
+every fixed star describes a circle of immense radius in the course of
+an astronomical day, a result which is opposed to the statement of the
+law of inertia. So that if we adhere to this law we must refer these
+motions only to systems of coordinates relative to which the fixed
+stars do not move in a circle. A system of co-ordinates of which the
+state of motion is such that the law of inertia holds relative to it is
+called a “Galileian system of co-ordinates.” The laws of the mechanics
+of Galilei-Newton can be regarded as valid only for a Galileian system
+of co-ordinates.
+
+
+V.
+
+THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)
+
+In order to attain the greatest possible clearness, let us return to
+our example of the railway carriage supposed to be travelling
+uniformly. We call its motion a uniform translation (“uniform” because
+it is of constant velocity and direction, “translation” because
+although the carriage changes its position relative to the embankment
+yet it does not rotate in so doing). Let us imagine a raven flying
+through the air in such a manner that its motion, as observed from the
+embankment, is uniform and in a straight line. If we were to observe
+the flying raven from the moving railway carriage. we should find that
+the motion of the raven would be one of different velocity and
+direction, but that it would still be uniform and in a straight line.
+Expressed in an abstract manner we may say: If a mass _m_ is moving
+uniformly in a straight line with respect to a co-ordinate system _K_,
+then it will also be moving uniformly and in a straight line relative
+to a second co-ordinate system _K′_ provided that the latter is
+executing a uniform translatory motion with respect to _K_. In
+accordance with the discussion contained in the preceding section, it
+follows that:
+
+If _K_ is a Galileian co-ordinate system. then every other co-ordinate
+system _K′_ is a Galileian one, when, in relation to _K_, it is in a
+condition of uniform motion of translation. Relative to _K′_ the
+mechanical laws of Galilei-Newton hold good exactly as they do with
+respect to _K_.
+
+We advance a step farther in our generalisation when we express the
+tenet thus: If, relative to _K_, _K′_ is a uniformly moving co-ordinate
+system devoid of rotation, then natural phenomena run their course with
+respect to _K′_ according to exactly the same general laws as with
+respect to _K_. This statement is called the _principle of relativity_
+(in the restricted sense).
+
+As long as one was convinced that all natural phenomena were capable of
+representation with the help of classical mechanics, there was no need
+to doubt the validity of this principle of relativity. But in view of
+the more recent development of electrodynamics and optics it became
+more and more evident that classical mechanics affords an insufficient
+foundation for the physical description of all natural phenomena. At
+this juncture the question of the validity of the principle of
+relativity became ripe for discussion, and it did not appear impossible
+that the answer to this question might be in the negative.
+
+Nevertheless, there are two general facts which at the outset speak
+very much in favour of the validity of the principle of relativity.
+Even though classical mechanics does not supply us with a sufficiently
+broad basis for the theoretical presentation of all physical phenomena,
+still we must grant it a considerable measure of “truth,” since it
+supplies us with the actual motions of the heavenly bodies with a
+delicacy of detail little short of wonderful. The principle of
+relativity must therefore apply with great accuracy in the domain of
+_mechanics_. But that a principle of such broad generality should hold
+with such exactness in one domain of phenomena, and yet should be
+invalid for another, is _a priori_ not very probable.
+
+We now proceed to the second argument, to which, moreover, we shall
+return later. If the principle of relativity (in the restricted sense)
+does not hold, then the Galileian co-ordinate systems _K, K′, K″_,
+etc., which are moving uniformly relative to each other, will not be
+_equivalent_ for the description of natural phenomena. In this case we
+should be constrained to believe that natural laws are capable of being
+formulated in a particularly simple manner, and of course only on
+condition that, from amongst all possible Galileian co-ordinate
+systems, we should have chosen _one_ (_K0_) of a particular state of
+motion as our body of reference. We should then be justified (because
+of its merits for the description of natural phenomena) in calling this
+system “absolutely at rest,” and all other Galileian systems _K_ “in
+motion.” If, for instance, our embankment were the system _K0_ then our
+railway carriage would be a system _K_, relative to which less simple
+laws would hold than with respect to _K0_. This diminished simplicity
+would be due to the fact that the carriage _K_ would be in motion
+(_i.e._ “really”)with respect to _K0_. In the general laws of nature
+which have been formulated with reference to _K_, the magnitude and
+direction of the velocity of the carriage would necessarily play a
+part. We should expect, for instance, that the note emitted by an
+organpipe placed with its axis parallel to the direction of travel
+would be different from that emitted if the axis of the pipe were
+placed perpendicular to this direction.
+
+Now in virtue of its motion in an orbit round the sun, our earth is
+comparable with a railway carriage travelling with a velocity of about
+30 kilometres per second. If the principle of relativity were not valid
+we should therefore expect that the direction of motion of the earth at
+any moment would enter into the laws of nature, and also that physical
+systems in their behaviour would be dependent on the orientation in
+space with respect to the earth. For owing to the alteration in
+direction of the velocity of revolution of the earth in the course of a
+year, the earth cannot be at rest relative to the hypothetical system
+_K0_ throughout the whole year. However, the most careful observations
+have never revealed such anisotropic properties in terrestrial physical
+space, _i.e._ a physical non-equivalence of different directions. This
+is very powerful argument in favour of the principle of relativity.
+
+
+VI.
+
+THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL
+MECHANICS
+
+Let us suppose our old friend the railway carriage to be travelling
+along the rails with a constant velocity _v_, and that a man traverses
+the length of the carriage in the direction of travel with a velocity
+_w_. How quickly or, in other words, with what velocity _W_ does the
+man advance relative to the embankment during the process? The only
+possible answer seems to result from the following consideration: If
+the man were to stand still for a second, he would advance relative to
+the embankment through a distance _v_ equal numerically to the velocity
+of the carriage. As a consequence of his walking, however, he traverses
+an additional distance w relative to the carriage, and hence also
+relative to the embankment, in this second, the distance w being
+numerically equal to the velocity with which he is walking. Thus in
+total he covers the distance _W = v + w_ relative to the embankment in
+the second considered. We shall see later that this result, which
+expresses the theorem of the addition of velocities employed in
+classical mechanics, cannot be maintained; in other words, the law that
+we have just written down does not hold in reality. For the time being,
+however, we shall assume its correctness.
+
+
+VII.
+
+THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH
+THE PRINCIPLE OF RELATIVITY
+
+There is hardly a simpler law in physics than that according to which
+light is propagated in empty space. Every child at school knows, or
+believes he knows, that this propagation takes place in straight lines
+with a velocity _c_ = 300,000 km./sec. At all events we know with great
+exactness that this velocity is the same for all colours, because if
+this were not the case, the minimum of emission would not be observed
+simultaneously for different colours during the eclipse of a fixed star
+by its dark neighbour. By means of similar considerations based on
+observations of double stars, the Dutch astronomer De Sitter was also
+able to show that the velocity of propagation of light cannot depend on
+the velocity of motion of the body emitting the light. The assumption
+that this velocity of propagation is dependent on the direction “in
+space” is in itself improbable.
+
+In short, let us assume that the simple law of the constancy of the
+velocity of light _c_ (in vacuum) is justifiably believed by the child
+at school. Who would imagine that this simple law has plunged the
+conscientiously thoughtful physicist into the greatest intellectual
+difficulties? Let us consider how these difficulties arise.
+
+Of course we must refer the process of the propagation of light (and
+indeed every other process) to a rigid reference-body (co-ordinate
+system). As such a system let us again choose our embankment. We shall
+imagine the air above it to have been removed. If a ray of light be
+sent along the embankment, we see from the above that the tip of the
+ray will be transmitted with the velocity _c_ relative to the
+embankment. Now let us suppose that our railway carriage is again
+travelling along the railway lines with the velocity _v_, and that its
+direction is the same as that of the ray of light, but its velocity of
+course much less. Let us inquire about the velocity of propagation of
+the ray of light relative to the carriage. It is obvious that we can
+here apply the consideration of the previous section, since the ray of
+light plays the part of the man walking along relatively to the
+carriage. The velocity _W_ of the man relative to the embankment is
+here replaced by the velocity of light relative to the embankment. _w_
+is the required velocity of light with respect to the carriage, and we
+have
+
+_w = c – v._
+
+The velocity of propagation ot a ray of light relative to the carriage
+thus comes out smaller than _c_.
+
+But this result comes into conflict with the principle of relativity
+set forth in Section V. For, like every other general law of nature,
+the law of the transmission of light _in vacuo_ [in vacuum] must,
+according to the principle of relativity, be the same for the railway
+carriage as reference-body as when the rails are the body of reference.
+But, from our above consideration, this would appear to be impossible.
+If every ray of light is propagated relative to the embankment with the
+velocity _c_, then for this reason it would appear that another law of
+propagation of light must necessarily hold with respect to the
+carriage—a result contradictory to the principle of relativity.
+
+In view of this dilemma there appears to be nothing else for it than to
+abandon either the principle of relativity or the simple law of the
+propagation of light _in vacuo_. Those of you who have carefully
+followed the preceding discussion are almost sure to expect that we
+should retain the principle of relativity, which appeals so
+convincingly to the intellect because it is so natural and simple. The
+law of the propagation of light _in vacuo_ would then have to be
+replaced by a more complicated law conformable to the principle of
+relativity. The development of theoretical physics shows, however, that
+we cannot pursue this course. The epoch-making theoretical
+investigations of H. A. Lorentz on the electrodynamical and optical
+phenomena connected with moving bodies show that experience in this
+domain leads conclusively to a theory of electromagnetic phenomena, of
+which the law of the constancy of the velocity of light in vacuo is a
+necessary consequence. Prominent theoretical physicists were therefore
+more inclined to reject the principle of relativity, in spite of the
+fact that no empirical data had been found which were contradictory to
+this principle.
+
+At this juncture the theory of relativity entered the arena. As a
+result of an analysis of the physical conceptions of time and space, it
+became evident that _in reality there is not the least incompatibilitiy
+between the principle of relativity and the law of propagation of
+light_, and that by systematically holding fast to both these laws a
+logically rigid theory could be arrived at. This theory has been called
+the _special theory of relativity_ to distinguish it from the extended
+theory, with which we shall deal later. In the following pages we shall
+present the fundamental ideas of the special theory of relativity.
+
+
+VIII.
+
+ON THE IDEA OF TIME IN PHYSICS
+
+Lightning has struck the rails on our railway embankment at two places
+_A_ and _B_ far distant from each other. I make the additional
+assertion that these two lightning flashes occurred simultaneously. If
+I ask you whether there is sense in this statement, you will answer my
+question with a decided “Yes.” But if I now approach you with the
+request to explain to me the sense of the statement more precisely, you
+find after some consideration that the answer to this question is not
+so easy as it appears at first sight.
+
+After some time perhaps the following answer would occur to you: “The
+significance of the statement is clear in itself and needs no further
+explanation; of course it would require some consideration if I were to
+be commissioned to determine by observations whether in the actual case
+the two events took place simultaneously or not.” I cannot be satisfied
+with this answer for the following reason. Supposing that as a result
+of ingenious considerations an able meteorologist were to discover that
+the lightning must always strike the places _A_ and _B_ simultaneously,
+then we should be faced with the task of testing whether or not this
+theoretical result is in accordance with the reality. We encounter the
+same difficulty with all physical statements in which the conception
+“simultaneous” plays a part. The concept does not exist for the
+physicist until he has the possibility of discovering whether or not it
+is fulfilled in an actual case. We thus require a definition of
+simultaneity such that this definition supplies us with the method by
+means of which, in the present case, he can decide by experiment
+whether or not both the lightning strokes occurred simultaneously. As
+long as this requirement is not satisfied, I allow myself to be
+deceived as a physicist (and of course the same applies if I am not a
+physicist), when I imagine that I am able to attach a meaning to the
+statement of simultaneity. (I would ask the reader not to proceed
+farther until he is fully convinced on this point.)
+
+After thinking the matter over for some time you then offer the
+following suggestion with which to test simultaneity. By measuring
+along the rails, the connecting line _AB_ should be measured up and an
+observer placed at the mid-point M of the distance _AB_. This observer
+should be supplied with an arrangement (_e.g._ two mirrors inclined at
+90°) which allows him visually to observe both places _A_ and _B_ at
+the same time. If the observer perceives the two flashes of lightning
+at the same time, then they are simultaneous.
+
+I am very pleased with this suggestion, but for all that I cannot
+regard the matter as quite settled, because I feel constrained to raise
+the following objection: “Your definition would certainly be right, if
+only I knew that the light by means of which the observer at _M_
+perceives the lightning flashes travels along the length _A_ → _M_ with
+the same velocity as along the length _B_ → _M_. But an examination of
+this supposition would only be possible if we already had at our
+disposal the means of measuring time. It would thus appear as though we
+were moving here in a logical circle.”
+
+After further consideration you cast a somewhat disdainful glance at
+me—and rightly so—and you declare: “I maintain my previous definition
+nevertheless, because in reality it assumes absolutely nothing about
+light. There is only _one_ demand to be made of the definition of
+simultaneity, namely, that in every real case it must supply us with an
+empirical decision as to whether or not the conception that has to be
+defined is fulfilled. That my definition satisfies this demand is
+indisputable. That light requires the same time to traverse the path
+_A_ → _M_ as for the path _B_ → _M_ is in reality neither a
+_supposition nor a hypothesis_ about the physical nature of light, but
+a _stipulation_ which I can make of my own freewill in order to arrive
+at a definition of simultaneity.”
+
+It is clear that this definition can be used to give an exact meaning
+not only to _two_ events, but to as many events as we care to choose,
+and independently of the positions of the scenes of the events with
+respect to the body of reference[7] (here the railway embankment). We
+are thus led also to a definition of “time” in physics. For this
+purpose we suppose that clocks of identical construction are placed at
+the points _A, B_ and _C_ of the railway line (co-ordinate system) and
+that they are set in such a manner that the positions of their pointers
+are simultaneously (in the above sense) the same. Under these
+conditions we understand by the “time” of an event the reading
+(position of the hands) of that one of these clocks which is in the
+immediate vicinity (in space) of the event. In this manner a time-value
+is associated with every event which is essentially capable of
+observation.
+
+
+ [7] We suppose further that, when three events _A, B_ and _C_ occur in
+ different places in such a manner that, if _A_ is simultaneous with
+ _B_, and _B_ is simultaneous with _C_ (simultaneous in the sense of
+ the above definition), then the criterion for the simultaneity of the
+ pair of events _A, C_ is also satisfied. This assumption is a physical
+ hypothesis about the law of propagation of light; it must certainly be
+ fulfilled if we are to maintain the law of the constancy of the
+ velocity of light _in vacuo_.
+
+
+This stipulation contains a further physical hypothesis, the validity
+of which will hardly be doubted without empirical evidence to the
+contrary. It has been assumed that all these clocks _go at the same
+rate_ if they are of identical construction. Stated more exactly: When
+two clocks arranged at rest in different places of a reference-body are
+set in such a manner that a _particular_ position of the pointers of
+the one clock is _simultaneous_ (in the above sense) with the _same_
+position, of the pointers of the other clock, then identical “settings”
+are always simultaneous (in the sense of the above definition).
+
+
+IX.
+
+THE RELATIVITY OF SIMULTANEITY
+
+Up to now our considerations have been referred to a particular body of
+reference, which we have styled a “railway embankment.” We suppose a
+very long train travelling along the rails with the constant velocity v
+and in the direction indicated in Fig 1. People travelling in this
+train will with a vantage view the train as a rigid reference-body
+(co-ordinate system); they regard all events in reference to the train.
+Then every event which takes place along the line also takes place at a
+particular point of the train. Also the definition of simultaneity can
+be given relative to the train in exactly the same way as with respect
+to the embankment. As a natural consequence, however, the following
+question arises:
+
+image001
+
+
+Are two events (_e.g._ the two strokes of lightning _A_ and _B_) which
+are simultaneous _with reference to the railway embankment_ also
+simultaneous _relatively to the train?_ We shall show directly that the
+answer must be in the negative.
+
+When we say that the lightning strokes _A_ and _B_ are simultaneous
+with respect to be embankment, we mean: the rays of light emitted at
+the places _A_ and _B_, where the lightning occurs, meet each other at
+the mid-point _M_ of the length _A_ → _B_ of the embankment. But the
+events _A_ and _B_ also correspond to positions _A_ and _B_ on the
+train. Let _M′_ be the mid-point of the distance _A_ → _B_ on the
+travelling train. Just when the flashes (as judged from the embankment)
+of lightning occur, this point _M′_ naturally coincides with the point
+_M_ but it moves towards the right in the diagram with the velocity v
+of the train. If an observer sitting in the position _M′_ in the train
+did not possess this velocity, then he would remain permanently at M,
+and the light rays emitted by the flashes of lightning _A_ and _B_
+would reach him simultaneously, _i.e._ they would meet just where he is
+situated. Now in reality (considered with reference to the railway
+embankment) he is hastening towards the beam of light coming from _B_,
+whilst he is riding on ahead of the beam of light coming from _A_.
+Hence the observer will see the beam of light emitted from _B_ earlier
+than he will see that emitted from _A_. Observers who take the railway
+train as their reference-body must therefore come to the conclusion
+that the lightning flash _B_ took place earlier than the lightning
+flash _A_. We thus arrive at the important result:
+
+Events which are simultaneous with reference to the embankment are not
+simultaneous with respect to the train, and _vice versa_ (relativity of
+simultaneity). Every reference-body (co-ordinate system) has its own
+particular time; unless we are told the reference-body to which the
+statement of time refers, there is no meaning in a statement of the
+time of an event.
+
+Now before the advent of the theory of relativity it had always tacitly
+been assumed in physics that the statement of time had an absolute
+significance, _i.e._ that it is independent of the state of motion of
+the body of reference. But we have just seen that this assumption is
+incompatible with the most natural definition of simultaneity; if we
+discard this assumption, then the conflict between the law of the
+propagation of light _in vacuo_ and the principle of relativity
+(developed in Section VII) disappears.
+
+We were led to that conflict by the considerations of Section VI, which
+are now no longer tenable. In that section we concluded that the man in
+the carriage, who traverses the distance _w per second_ relative to the
+carriage, traverses the same distance also with respect to the
+embankment _in each second_ of time. But, according to the foregoing
+considerations, the time required by a particular occurrence with
+respect to the carriage must not be considered equal to the duration of
+the same occurrence as judged from the embankment (as reference-body).
+Hence it cannot be contended that the man in walking travels the
+distance _w_ relative to the railway line in a time which is equal to
+one second as judged from the embankment.
+
+Moreover, the considerations of Section VI are based on yet a second
+assumption, which, in the light of a strict consideration, appears to
+be arbitrary, although it was always tacitly made even before the
+introduction of the theory of relativity.
+
+
+X.
+
+ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
+
+Let us consider two particular points on the train [8] travelling along
+the embankment with the velocity _v_, and inquire as to their distance
+apart. We already know that it is necessary to have a body of reference
+for the measurement of a distance, with respect to which body the
+distance can be measured up. It is the simplest plan to use the train
+itself as reference-body (co-ordinate system). An observer in the train
+measures the interval by marking off his measuring-rod in a straight
+line (_e.g._ along the floor of the carriage) as many times as is
+necessary to take him from the one marked point to the other. Then the
+number which tells us how often the rod has to be laid down is the
+required distance.
+
+
+ [8] _e.g._ the middle of the first and of the hundredth carriage.
+
+
+It is a different matter when the distance has to be judged from the
+railway line. Here the following method suggests itself. If we call
+_A′_ and _B′_ the two points on the train whose distance apart is
+required, then both of these points are moving with the velocity v
+along the embankment. In the first place we require to determine the
+points _A_ and _B_ of the embankment which are just being passed by the
+two points _A′_ and _B′_ at a particular time t—judged from the
+embankment. These points _A_ and _B_ of the embankment can be
+determined by applying the definition of time given in Section VIII.
+The distance between these points A and B is then measured by repeated
+application of the measuring-rod along the embankment.
+
+_A priori_ it is by no means certain that this last measurement will
+supply us with the same result as the first. Thus the length of the
+train as measured from the embankment may be different from that
+obtained by measuring in the train itself. This circumstance leads us
+to a second objection which must be raised against the apparently
+obvious consideration of Section VI. Namely, if the man in the carriage
+covers the distance _w_ in a unit of time—_measured from the
+train_,—then this distance—_as measured from the embankment_ is not
+necessarily also equal to _w_.
+
+
+XI.
+
+THE LORENTZ TRANSFORMATION
+
+The results of the last three sections show that the apparent
+incompatibility of the law of propagation of light with the principle
+of relativity (Section VII) has been derived by means of a
+consideration which borrowed two unjustifiable hypotheses from
+classical mechanics; these are as follows:
+
+(1) The time-interval (time) between two events is independent of the
+condition of motion of the body of reference.
+
+
+(2) The space-interval (distance) between two points of a rigid body is
+independent of the condition of motion of the body of reference.
+
+
+If we drop these hypotheses, then the dilemma of Section VII
+disappears, because the theorem of the addition of velocities derived
+in Section VI becomes invalid. The possibility presents itself that the
+law of the propagation of light _in vacuo_ may be compatible with the
+principle of relativity, and the question arises: How have we to modify
+the considerations of Section VI in order to remove the apparent
+disagreement between these two fundamental results of experience? This
+question leads to a general one. In the discussion of Section VI we
+have to do with places and times relative both to the train and to the
+embankment. How are we to find the place and time of an event in
+relation to the train, when we know the place and time of the event
+with respect to the railway embankment? Is there a thinkable answer to
+this question of such a nature that the law of transmission of light
+_in vacuo_ does not contradict the principle of relativity? In other
+words: Can we conceive of a relation between place and time of the
+individual events relative to both reference-bodies, such that every
+ray of light possesses the velocity of transmission _c_ relative to the
+embankment and relative to the train? This question leads to a quite
+definite positive answer, and to a perfectly definite transformation
+law for the space-time magnitudes of an event when changing over from
+one body of reference to another.
+
+Before we deal with this, we shall introduce the following incidental
+consideration. Up to the present we have only considered events taking
+place along the embankment, which had mathematically to assume the
+function of a straight line. In the manner indicated in Section II we
+can imagine this reference-body supplemented laterally and in a
+vertical direction by means of a framework of rods, so that an event
+which takes place anywhere can be localised with reference to this
+framework. Similarly, we can imagine the train travelling with the
+velocity _v_ to be continued across the whole of space, so that every
+event, no matter how far off it may be, could also be localised with
+respect to the second framework. Without committing any fundamental
+error, we can disregard the fact that in reality these frameworks would
+continually interfere with each other, owing to the impenetrability of
+solid bodies. In every such framework we imagine three surfaces
+perpendicular to each other marked out, and designated as “co-ordinate
+planes” (“co-ordinate system”). A co-ordinate system _K_ then
+corresponds to the embankment, and a co-ordinate system _K′_ to the
+train. An event, wherever it may have taken place, would be fixed in
+space with respect to _K_ by the three perpendiculars _x, y, z_ on the
+co-ordinate planes, and with regard to time by a time value _t_.
+Relative to _K′, the same event_ would be fixed in respect of space and
+time by corresponding values _x′, y′, z′, t′_, which of course are not
+identical with _x, y, z, t_. It has already been set forth in detail
+how these magnitudes are to be regarded as results of physical
+measurements.
+
+image002
+
+
+Obviously our problem can be exactly formulated in the following
+manner. What are the values _x′, y′, z′, t′_, of an event with respect
+to _K′_, when the magnitudes _x, y, z, t_, of the same event with
+respect to _K_ are given? The relations must be so chosen that the law
+of the transmission of light in vacuo is satisfied for one and the same
+ray of light (and of course for every ray) with respect to _K_ and
+_K′_. For the relative orientation in space of the co-ordinate systems
+indicated in the diagram (Fig. 2), this problem is solved by means of
+the equations:
+
+image003
+
+
+_y′_ = _y_
+
+_z′_ = _z_
+
+
+image004
+
+
+This system of equations is known as the “Lorentz transformation.”[9]
+
+
+ [9] A simple derivation of the Lorentz transformation is given in
+ Appendix I.
+
+
+If in place of the law of transmission of light we had taken as our
+basis the tacit assumptions of the older mechanics as to the absolute
+character of times and lengths, then instead of the above we should
+have obtained the following equations:
+
+_x′_ = _x_ – _vt_
+
+
+_y′_ = _y_
+
+
+_z′_ = _z_
+
+
+_t′_ = _t_
+
+
+This system of equations is often termed the “Galilei transformation.”
+The Galilei transformation can be obtained from the Lorentz
+transformation by substituting an infinitely large value for the
+velocity of light _c_ in the latter transformation.
+
+Aided by the following illustration, we can readily see that, in
+accordance with the Lorentz transformation, the law of the transmission
+of light _in vacuo_ is satisfied both for the reference-body _K_ and
+for the reference-body _K′_. A light-signal is sent along the positive
+_x_-axis, and this light-stimulus advances in accordance with the
+equation
+
+_x_ = _ct_,
+
+
+_i.e._ with the velocity _c_. According to the equations of the Lorentz
+transformation, this simple relation between _x_ and _t_ involves a
+relation between _x′_ and _t′_. In point of fact, if we substitute for
+_x_ the value _ct_ in the first and fourth equations of the Lorentz
+transformation, we obtain:
+
+image005
+
+
+from which, by division, the expression
+
+_x′_ = _ct′_
+
+
+immediately follows. If referred to the system _K′_, the propagation of
+light takes place according to this equation. We thus see that the
+velocity of transmission relative to the reference-body _K′_ is also
+equal to _c_. The same result is obtained for rays of light advancing
+in any other direction whatsoever. Of cause this is not surprising,
+since the equations of the Lorentz transformation were derived
+conformably to this point of view.
+
+
+XII.
+
+THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
+
+Place a metre-rod in the _x′_-axis of _K′_ in such a manner that one
+end (the beginning) coincides with the point _x′_ = 0 whilst the other
+end (the end of the rod) coincides with the point _x′_ = 1. What is the
+length of the metre-rod relatively to the system _K_? In order to learn
+this, we need only ask where the beginning of the rod and the end of
+the rod lie with respect to _K_ at a particular time _t_ of the system
+_K_. By means of the first equation of the Lorentz transformation the
+values of these two points at the time _t_ = 0 can be shown to be
+
+image006
+
+
+the distance between the points being
+
+image007
+
+
+But the metre-rod is moving with the velocity _v_ relative to _K_. It
+therefore follows that the length of a rigid metre-rod moving in the
+direction of its length with a velocity _v_ is
+
+image008
+
+
+of a metre. The rigid rod is thus shorter when in motion than when at
+rest, and the more quickly it is moving, the shorter is the rod. For
+the velocity _v_ = _c_ we should have
+
+image009
+
+
+and for still greater velocities the square-root becomes imaginary.
+From this we conclude that in the theory of relativity the velocity _c_
+plays the part of a limiting velocity, which can neither be reached nor
+exceeded by any real body.
+
+Of course this feature of the velocity _c_ as a limiting velocity also
+clearly follows from the equations of the Lorentz transformation, for
+these became meaningless if we choose values of _v_ greater than _c_.
+
+If, on the contrary, we had considered a metre-rod at rest in the
+_x_-axis with respect to _K_, then we should have found that the length
+of the rod as judged from _K′_ would have been
+
+image010
+
+
+this is quite in accordance with the principle of relativity which
+forms the basis of our considerations.
+
+_A priori_ it is quite clear that we must be able to learn something
+about the physical behaviour of measuring-rods and clocks from the
+equations of transformation, for the magnitudes _z, y, x, t_, are
+nothing more nor less than the results of measurements obtainable by
+means of measuring-rods and clocks. If we had based our considerations
+on the Galileian transformation we should not have obtained a
+contraction of the rod as a consequence of its motion.
+
+Let us now consider a seconds-clock which is permanently situated at
+the origin (_x′_ = 0) of _K′_. _t′_ = 0 and _t′_ = 1 are two successive
+ticks of this clock. The first and fourth equations of the Lorentz
+transformation give for these two ticks:
+
+_t_ = 0
+
+and
+
+image011
+
+
+As judged from _K_, the clock is moving with the velocity _v_; as
+judged from this reference-body, the time which elapses between two
+strokes of the clock is not one second, but
+
+image012
+
+
+seconds, _i.e._ a somewhat larger time. As a consequence of its motion
+the clock goes more slowly than when at rest. Here also the velocity
+_c_ plays the part of an unattainable limiting velocity.
+
+
+XIII.
+
+THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU
+
+Now in practice we can move clocks and measuring-rods only with
+velocities that are small compared with the velocity of light; hence we
+shall hardly be able to compare the results of the previous section
+directly with the reality. But, on the other hand, these results must
+strike you as being very singular, and for that reason I shall now draw
+another conclusion from the theory, one which can easily be derived
+from the foregoing considerations, and which has been most elegantly
+confirmed by experiment.
+
+In Section VI we derived the theorem of the addition of velocities in
+one direction in the form which also results from the hypotheses of
+classical mechanics. This theorem can also be deduced readily from the
+Galilei transformation (Section XI). In place of the man walking inside
+the carriage, we introduce a point moving relatively to the co-ordinate
+system _K′_ in accordance with the equation
+
+_x′_ = _wt′_
+
+By means of the first and fourth equations of the Galilei
+transformation we can express _x′_ and _t′_ in terms of _x_ and _t_,
+and we then obtain
+
+_x_ = (_v_ + _w_)_t_
+
+This equation expresses nothing else than the law of motion of the
+point with reference to the system _K_ (of the man with reference to
+the embankment). We denote this velocity by the symbol _W_, and we then
+obtain, as in Section VI,
+
+_W_ = _v_ + _w_ . . . . . . . (A).
+
+But we can carry out this consideration just as well on the basis of
+the theory of relativity. In the equation
+
+_x′_ = _wt′_
+
+we must then express _x′_ and _t′_ in terms of _x_ and _t_, making use
+of the first and fourth equations of the _Lorentz transformation_.
+Instead of the equation (A) we then obtain the equation
+
+image013
+
+
+which corresponds to the theorem of addition for velocities in one
+direction according to the theory of relativity. The question now
+arises as to which of these two theorems is the better in accord with
+experience. On this point we are enlightened by a most important
+experiment which the brilliant physicist Fizeau performed more than
+half a century ago, and which has been repeated since then by some of
+the best experimental physicists, so that there can be no doubt about
+its result. The experiment is concerned with the following question.
+Light travels in a motionless liquid with a particular velocity _w_.
+How quickly does it travel in the direction of the arrow in the tube
+_T_ (see the accompanying diagram, Fig. 3) when the liquid above
+mentioned is flowing through the tube with a velocity _v_?
+
+image014
+
+
+In accordance with the principle of relativity we shall certainly have
+to take for granted that the propagation of light always takes place
+with the same velocity _w with respect to the liquid_, whether the
+latter is in motion with reference to other bodies or not. The velocity
+of light relative to the liquid and the velocity of the latter relative
+to the tube are thus known, and we require the velocity of light
+relative to the tube.
+
+It is clear that we have the problem of Section VI again before us. The
+tube plays the part of the railway embankment or of the co-ordinate
+system _K_, the liquid plays the part of the carriage or of the
+co-ordinate system _K′_, and finally, the light plays the part of the
+man walking along the carriage, or of the moving point in the present
+section. If we denote the velocity of the light relative to the tube by
+_W_, then this is given by the equation (A) or (B), according as the
+Galilei transformation or the Lorentz transformation corresponds to the
+facts. Experiment[10] decides in favour of equation (B) derived from
+the theory of relativity, and the agreement is, indeed, very exact.
+According to recent and most excellent measurements by Zeeman, the
+influence of the velocity of flow _v_ on the propagation of light is
+represented by formula (B) to within one per cent.
+
+
+ [10] Fizeau found
+
+
+image015
+
+
+where
+
+
+image016
+
+
+is the index of refraction of the liquid. On the other hand, owing to
+the smallness of
+
+
+image017
+
+
+as compared with 1, we can replace (B) in the first place by
+
+
+image018
+
+
+or to the same order of approximation by
+
+
+image019
+
+
+which agrees with Fizeau’s result.
+
+
+Nevertheless we must now draw attention to the fact that a theory of
+this phenomenon was given by H. A. Lorentz long before the statement of
+the theory of relativity. This theory was of a purely electrodynamical
+nature, and was obtained by the use of particular hypotheses as to the
+electromagnetic structure of matter. This circumstance, however, does
+not in the least diminish the conclusiveness of the experiment as a
+crucial test in favour of the theory of relativity, for the
+electrodynamics of Maxwell-Lorentz, on which the original theory was
+based, in no way opposes the theory of relativity. Rather has the
+latter been developed trom electrodynamics as an astoundingly simple
+combination and generalisation of the hypotheses, formerly independent
+of each other, on which electrodynamics was built.
+
+
+XIV.
+
+THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
+
+Our train of thought in the foregoing pages can be epitomised in the
+following manner. Experience has led to the conviction that, on the one
+hand, the principle of relativity holds true and that on the other hand
+the velocity of transmission of light _in vacuo_ has to be considered
+equal to a constant _c_. By uniting these two postulates we obtained
+the law of transformation for the rectangular co-ordinates _x, y, z_
+and the time _t_ of the events which constitute the processes of
+nature. In this connection we did not obtain the Galilei
+transformation, but, differing from classical mechanics, the _Lorentz
+transformation_.
+
+The law of transmission of light, the acceptance of which is justified
+by our actual knowledge, played an important part in this process of
+thought. Once in possession of the Lorentz transformation, however, we
+can combine this with the principle of relativity, and sum up the
+theory thus:
+
+Every general law of nature must be so constituted that it is
+transformed into a law of exactly the same form when, instead of the
+space-time variables _x, y, z, t_ of the original coordinate system
+_K_, we introduce new space-time variables _x′, y′, z′, t′_ of a
+co-ordinate system _K′_. In this connection the relation between the
+ordinary and the accented magnitudes is given by the Lorentz
+transformation. Or in brief: General laws of nature are co-variant with
+respect to Lorentz transformations.
+
+This is a definite mathematical condition that the theory of relativity
+demands of a natural law, and in virtue of this, the theory becomes a
+valuable heuristic aid in the search for general laws of nature. If a
+general law of nature were to be found which did not satisfy this
+condition, then at least one of the two fundamental assumptions of the
+theory would have been disproved. Let us now examine what general
+results the latter theory has hitherto evinced.
+
+
+XV.
+
+GENERAL RESULTS OF THE THEORY
+
+It is clear from our previous considerations that the (special) theory
+of relativity has grown out of electrodynamics and optics. In these
+fields it has not appreciably altered the predictions of theory, but it
+has considerably simplified the theoretical structure, _i.e._ the
+derivation of laws, and—what is incomparably more important—it has
+considerably reduced the number of independent hypotheses forming the
+basis of theory. The special theory of relativity has rendered the
+Maxwell-Lorentz theory so plausible, that the latter would have been
+generally accepted by physicists even if experiment had decided less
+unequivocally in its favour.
+
+Classical mechanics required to be modified before it could come into
+line with the demands of the special theory of relativity. For the main
+part, however, this modification affects only the laws for rapid
+motions, in which the velocities of matter _v_ are not very small as
+compared with the velocity of light. We have experience of such rapid
+motions only in the case of electrons and ions; for other motions the
+variations from the laws of classical mechanics are too small to make
+themselves evident in practice. We shall not consider the motion of
+stars until we come to speak of the general theory of relativity. In
+accordance with the theory of relativity the kinetic energy of a
+material point of mass _m_ is no longer given by the well-known
+expression
+
+image020
+
+
+but by the expression
+
+image021
+
+
+This expression approaches infinity as the velocity _v_ approaches the
+velocity of light _c_. The velocity must therefore always remain less
+than _c_, however great may be the energies used to produce the
+acceleration. If we develop the expression for the kinetic energy in
+the form of a series, we obtain
+
+image022
+
+
+When
+
+image023
+
+
+is small compared with unity, the third of these terms is always small
+in comparison with the second, which last is alone considered in
+classical mechanics. The first term _mc_2 does not contain the
+velocity, and requires no consideration if we are only dealing with the
+question as to how the energy of a point-mass; depends on the velocity.
+We shall speak of its essential significance later.
+
+The most important result of a general character to which the special
+theory of relativity has led is concerned with the conception of mass.
+Before the advent of relativity, physics recognised two conservation
+laws of fundamental importance, namely, the law of the conservation of
+energy and the law of the conservation of mass these two fundamental
+laws appeared to be quite independent of each other. By means of the
+theory of relativity they have been united into one law. We shall now
+briefly consider how this unification came about, and what meaning is
+to be attached to it.
+
+The principle of relativity requires that the law of the conservation
+of energy should hold not only with reference to a co-ordinate system
+_K_, but also with respect to every co-ordinate system _K′_ which is in
+a state of uniform motion of translation relative to _K_, or, briefly,
+relative to every “Galileian” system of co-ordinates. In contrast to
+classical mechanics; the Lorentz transformation is the deciding factor
+in the transition from one such system to another.
+
+By means of comparatively simple considerations we are led to draw the
+following conclusion from these premises, in conjunction with the
+fundamental equations of the electrodynamics of Maxwell: A body moving
+with the velocity _v_, which absorbs[11] an amount of energy _E_0 in
+the form of radiation without suffering an alteration in velocity in
+the process, has, as a consequence, its energy increased by an amount
+
+image024
+
+
+
+ [11] _E_0 is the energy taken up, as judged from a co-ordinate system
+ moving with the body.
+
+
+In consideration of the expression given above for the kinetic energy
+of the body, the required energy of the body comes out to be
+
+image025
+
+
+Thus the body has the same energy as a body of mass
+
+image026
+
+
+moving with the velocity _v_. Hence we can say: If a body takes up an
+amount of energy _E_0, then its inertial mass increases by an amount
+
+image027
+
+
+the inertial mass of a body is not a constant but varies according to
+the change in the energy of the body. The inertial mass of a system of
+bodies can even be regarded as a measure of its energy. The law of the
+conservation of the mass of a system becomes identical with the law of
+the conservation of energy, and is only valid provided that the system
+neither takes up nor sends out energy. Writing the expression for the
+energy in the form
+
+image028
+
+
+we see that the term _mc_2, which has hitherto attracted our attention,
+is nothing else than the energy possessed by the body[12] before it
+absorbed the energy _E_0.
+
+
+ [12] As judged from a co-ordinate system moving with the body.
+
+
+A direct comparison of this relation with experiment is not possible at
+the present time (1920; see[Note], p. 48), owing to the fact that the
+changes in energy _E_0 to which we can subject a system are not large
+enough to make themselves perceptible as a change in the inertial mass
+of the system.
+
+image027
+
+
+is too small in comparison with the mass _m_, which was present before
+the alteration of the energy. It is owing to this circumstance that
+classical mechanics was able to establish successfully the conservation
+of mass as a law of independent validity.
+
+
+ [Note] The equation E = mc2 has been thoroughly proved time and again
+ since this time.
+
+
+Let me add a final remark of a fundamental nature. The success of the
+Faraday-Maxwell interpretation of electromagnetic action at a distance
+resulted in physicists becoming convinced that there are no such things
+as instantaneous actions at a distance (not involving an intermediary
+medium) of the type of Newton’s law of gravitation.
+
+According to the theory of relativity, action at a distance with the
+velocity of light always takes the place of instantaneous action at a
+distance or of action at a distance with an infinite velocity of
+transmission. This is connected with the fact that the velocity _c_
+plays a fundamental role in this theory. In Part II we shall see in
+what way this result becomes modified in the general theory of
+relativity.
+
+
+XVI.
+
+EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
+
+To what extent is the special theory of relativity supported by
+experience? This question is not easily answered for the reason already
+mentioned in connection with the fundamental experiment of Fizeau. The
+special theory of relativity has crystallised out from the
+Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
+experience which support the electromagnetic theory also support the
+theory of relativity. As being of particular importance, I mention here
+the fact that the theory of relativity enables us to predict the
+effects produced on the light reaching us from the fixed stars. These
+results are obtained in an exceedingly simple manner, and the effects
+indicated, which are due to the relative motion of the earth with
+reference to those fixed stars are found to be in accord with
+experience. We refer to the yearly movement of the apparent position of
+the fixed stars resulting from the motion of the earth round the sun
+(aberration), and to the influence of the radial components of the
+relative motions of the fixed stars with respect to the earth on the
+colour of the light reaching us from them. The latter effect manifests
+itself in a slight displacement of the spectral lines of the light
+transmitted to us from a fixed star, as compared with the position of
+the same spectral lines when they are produced by a terrestrial source
+of light (Doppler principle). The experimental arguments in favour of
+the Maxwell-Lorentz theory, which are at the same time arguments in
+favour of the theory of relativity, are too numerous to be set forth
+here. In reality they limit the theoretical possibilities to such an
+extent, that no other theory than that of Maxwell and Lorentz has been
+able to hold its own when tested by experience.
+
+But there are two classes of experimental facts hitherto obtained which
+can be represented in the Maxwell-Lorentz theory only by the
+introduction of an auxiliary hypothesis, which in itself—_i.e._ without
+making use of the theory of relativity—appears extraneous.
+
+It is known that cathode rays and the so-called β-rays emitted by
+radioactive substances consist of negatively electrified particles
+(electrons) of very small inertia and large velocity. By examining the
+deflection of these rays under the influence of electric and magnetic
+fields, we can study the law of motion of these particles very exactly.
+
+In the theoretical treatment of these electrons, we are faced with the
+difficulty that electrodynamic theory of itself is unable to give an
+account of their nature. For since electrical masses of one sign repel
+each other, the negative electrical masses constituting the electron
+would necessarily be scattered under the influence of their mutual
+repulsions, unless there are forces of another kind operating between
+them, the nature of which has hitherto remained obscure to us.[13] If
+we now assume that the relative distances between the electrical masses
+constituting the electron remain unchanged during the motion of the
+electron (rigid connection in the sense of classical mechanics), we
+arrive at a law of motion of the electron which does not agree with
+experience. Guided by purely formal points of view, H. A. Lorentz was
+the first to introduce the hypothesis that the form of the electron
+experiences a contraction in the direction of motion in consequence of
+that motion. the contracted length being proportional to the expression
+
+image029
+
+
+This, hypothesis, which is not justifiable by any electrodynamical
+facts, supplies us then with that particular law of motion which has
+been confirmed with great precision in recent years.
+
+
+ [13] The general theory of relativity renders it likely that the
+ electrical masses of an electron are held together by gravitational
+ forces.
+
+
+The theory of relativity leads to the same law of motion, without
+requiring any special hypothesis whatsoever as to the structure and the
+behaviour of the electron. We arrived at a similar conclusion in
+Section XIII in connection with the experiment of Fizeau, the result of
+which is foretold by the theory of relativity without the necessity of
+drawing on hypotheses as to the physical nature of the liquid.
+
+The second class of facts to which we have alluded has reference to the
+question whether or not the motion of the earth in space can be made
+perceptible in terrestrial experiments. We have already remarked in
+Section V that all attempts of this nature led to a negative result.
+Before the theory of relativity was put forward, it was difficult to
+become reconciled to this negative result, for reasons now to be
+discussed. The inherited prejudices about time and space did not allow
+any doubt to arise as to the prime importance of the Galileian
+transformation for changing over from one body of reference to another.
+Now assuming that the Maxwell-Lorentz equations hold for a
+reference-body _K_, we then find that they do not hold for a
+reference-body _K′_ moving uniformly with respect to _K_, if we assume
+that the relations of the Galileian transformation exist between the
+co-ordinates of _K_ and _K′_. It thus appears that, of all Galileian
+co-ordinate systems, one (_K_) corresponding to a particular state of
+motion is physically unique. This result was interpreted physically by
+regarding _K_ as at rest with respect to a hypothetical æther of space.
+On the other hand, all coordinate systems _K′_ moving relatively to _K_
+were to be regarded as in motion with respect to the æther. To this
+motion of _K′_ against the æther (“æther-drift” relative to _K′_) were
+attributed the more complicated laws which were supposed to hold
+relative to _K′_. Strictly speaking, such an æther-drift ought also to
+be assumed relative to the earth, and for a long time the efforts of
+physicists were devoted to attempts to detect the existence of an
+æther-drift at the earth’s surface.
+
+In one of the most notable of these attempts Michelson devised a method
+which appears as though it must be decisive. Imagine two mirrors so
+arranged on a rigid body that the reflecting surfaces face each other.
+A ray of light requires a perfectly definite time _T_ to pass from one
+mirror to the other and back again, if the whole system be at rest with
+respect to the æther. It is found by calculation, however, that a
+slightly different time _T′_ is required for this process, if the body,
+together with the mirrors, be moving relatively to the æther. And yet
+another point: it is shown by calculation that for a given velocity _v_
+with reference to the æther, this time _T′_ is different when the body
+is moving perpendicularly to the planes of the mirrors from that
+resulting when the motion is parallel to these planes. Although the
+estimated difference between these two times is exceedingly small,
+Michelson and Morley performed an experiment involving interference in
+which this difference should have been clearly detectable. But the
+experiment gave a negative result—a fact very perplexing to physicists.
+Lorentz and FitzGerald rescued the theory from this difficulty by
+assuming that the motion of the body relative to the æther produces a
+contraction of the body in the direction of motion, the amount of
+contraction being just sufficient to compensate for the difference in
+time mentioned above. Comparison with the discussion in Section XII
+shows that also from the standpoint of the theory of relativity this
+solution of the difficulty was the right one. But on the basis of the
+theory of relativity the method of interpretation is incomparably more
+satisfactory. According to this theory there is no such thing as a
+“specially favoured” (unique) co-ordinate system to occasion the
+introduction of the æther-idea, and hence there can be no æther-drift,
+nor any experiment with which to demonstrate it. Here the contraction
+of moving bodies follows from the two fundamental principles of the
+theory, without the introduction of particular hypotheses; and as the
+prime factor involved in this contraction we find, not the motion in
+itself, to which we cannot attach any meaning, but the motion with
+respect to the body of reference chosen in the particular case in
+point. Thus for a co-ordinate system moving with the earth the mirror
+system of Michelson and Morley is not shortened, but it _is_ shortened
+for a co-ordinate system which is at rest relatively to the sun.
+
+
+XVII.
+
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE
+
+The non-mathematician is seized by a mysterious shuddering when he
+hears of “four-dimensional” things, by a feeling not unlike that
+awakened by thoughts of the occult. And yet there is no more
+common-place statement than that the world in which we live is a
+four-dimensional space-time continuum.
+
+Space is a three-dimensional continuum. By this we mean that it is
+possible to describe the position of a point (at rest) by means of
+three numbers (co-ordinates) _x, y, z_, and that there is an indefinite
+number of points in the neighbourhood of this one, the position of
+which can be described by co-ordinates such as _x1, y1, z1_, which may
+be as near as we choose to the respective values of the co-ordinates
+_x, y, z_, of the first point. In virtue of the latter property we
+speak of a “continuum,” and owing to the fact that there are three
+co-ordinates we speak of it as being “three-dimensional.”
+
+Similarly, the world of physical phenomena which was briefly called
+“world” by Minkowski is naturally four dimensional in the space-time
+sense. For it is composed of individual events, each of which is
+described by four numbers, namely, three space co-ordinates _x, y, z_,
+and a time co-ordinate, the time value _t_. The “world” is in this
+sense also a continuum; for to every event there are as many
+“neighbouring” events (realised or at least thinkable) as we care to
+choose, the co-ordinates _x1, y1, z1, t1_ of which differ by an
+indefinitely small amount from those of the event _x, y, z, t_
+originally considered. That we have not been accustomed to regard the
+world in this sense as a four-dimensional continuum is due to the fact
+that in physics, before the advent of the theory of relativity, time
+played a different and more independent rôle, as compared with the
+space coordinates. It is for this reason that we have been in the habit
+of treating time as an independent continuum. As a matter of fact,
+according to classical mechanics, time is absolute, _i.e._ it is
+independent of the position and the condition of motion of the system
+of co-ordinates. We see this expressed in the last equation of the
+Galileian transformation (_t′_ = _t_).
+
+The four-dimensional mode of consideration of the “world” is natural on
+the theory of relativity, since according to this theory time is robbed
+of its independence. This is shown by the fourth equation of the
+Lorentz transformation:
+
+image030
+
+
+Moreover, according to this equation the time difference Δ_t′_ of two
+events with respect to _K′_ does not in general vanish, even when the
+time difference Δ_t_ of the same events with reference to _K_ vanishes.
+Pure “space-distance” of two events with respect to _K_ results in
+“time-distance ” of the same events with respect to _K_. But the
+discovery of Minkowski, which was of importance for the formal
+development of the theory of relativity, does not lie here. It is to be
+found rather in the fact of his recognition that the four-dimensional
+space-time continuum of the theory of relativity, in its most essential
+formal properties, shows a pronounced relationship to the
+three-dimensional continuum of Euclidean geometrical space.[14] In
+order to give due prominence to this relationship, however, we must
+replace the usual time co-ordinate t by an imaginary magnitude
+
+image031
+
+
+proportional to it. Under these conditions, the natural laws satisfying
+the demands of the (special) theory of relativity assume mathematical
+forms, in which the time co-ordinate plays exactly the same role as the
+three space co-ordinates. Formally, these four co-ordinates correspond
+exactly to the three space co-ordinates in Euclidean geometry. It must
+be clear even to the non-mathematician that, as a consequence of this
+purely formal addition to our knowledge, the theory perforce gained
+clearness in no mean measure.
+
+
+ [14] Cf. the somewhat more detailed discussion in Appendix II.
+
+
+These inadequate remarks can give the reader only a vague notion of the
+important idea contributed by Minkowski. Without it the general theory
+of relativity, of which the fundamental ideas are developed in the
+following pages, would perhaps have got no farther than its long
+clothes. Minkowski’s work is doubtless difficult of access to anyone
+inexperienced in mathematics, but since it is not necessary to have a
+very exact grasp of this work in order to understand the fundamental
+ideas of either the special or the general theory of relativity, I
+shall leave it here at present, and revert to it only towards the end
+of Part II.
+
+
+PART II: THE GENERAL THEORY OF RELATIVITY
+
+
+XVIII.
+
+SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
+
+The basal principle, which was the pivot of all our previous
+considerations, was the _special_ principle of relativity, _i.e._ the
+principle of the physical relativity of all _uniform_ motion. Let as
+once more analyse its meaning carefully.
+
+It was at all times clear that, from the point of view of the idea it
+conveys to us, every motion must be considered only as a relative
+motion. Returning to the illustration we have frequently used of the
+embankment and the railway carriage, we can express the fact of the
+motion here taking place in the following two forms, both of which are
+equally justifiable:
+
+(_a_) The carriage is in motion relative to the embankment,
+
+
+(_b_) The embankment is in motion relative to the carriage.
+
+
+In (_a_) the embankment, in (_b_) the carriage, serves as the body of
+reference in our statement of the motion taking place. If it is simply
+a question of detecting or of describing the motion involved, it is in
+principle immaterial to what reference-body we refer the motion. As
+already mentioned, this is self-evident, but it must not be confused
+with the much more comprehensive statement called “the principle of
+relativity,” which we have taken as the basis of our investigations.
+
+The principle we have made use of not only maintains that we may
+equally well choose the carriage or the embankment as our
+reference-body for the description of any event (for this, too, is
+self-evident). Our principle rather asserts what follows: If we
+formulate the general laws of nature as they are obtained from
+experience, by making use of
+
+(_a_) the embankment as reference-body,
+
+
+(_b_) the railway carriage as reference-body,
+
+
+then these general laws of nature (_e.g._ the laws of mechanics or the
+law of the propagation of light _in vacuo_) have exactly the same form
+in both cases. This can also be expressed as follows: For the physical
+description of natural processes, neither of the reference bodies _K,
+K′_ is unique (lit. “specially marked out”) as compared with the other.
+Unlike the first, this latter statement need not of necessity hold _a
+priori;_ it is not contained in the conceptions of “motion” and
+“reference-body” and derivable from them; only _experience_ can decide
+as to its correctness or incorrectness.
+
+Up to the present, however, we have by no means maintained the
+equivalence of _all_ bodies of reference _K_ in connection with the
+formulation of natural laws. Our course was more on the following
+Iines. In the first place, we started out from the assumption that
+there exists a reference-body _K_, whose condition of motion is such
+that the Galileian law holds with respect to it: A particle left to
+itself and sufficiently far removed from all other particles moves
+uniformly in a straight line. With reference to K (Galileian
+reference-body) the laws of nature were to be as simple as possible.
+But in addition to K, all bodies of reference _K′_ should be given
+preference in this sense, and they should be exactly equivalent to _K_
+for the formulation of natural laws, provided that they are in a state
+of _uniform rectilinear and non-rotary motion_ with respect to _K_; all
+these bodies of reference are to be regarded as Galileian
+reference-bodies. The validity of the principle of relativity was
+assumed only for these reference-bodies, but not for others (_e.g._
+those possessing motion of a different kind). In this sense we speak of
+the _special_ principle of relativity, or special theory of relativity.
+
+In contrast to this we wish to understand by the “general principle of
+relativity” the following statement: All bodies of reference _K, K′_,
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion. But before proceeding farther, it ought to be pointed
+out that this formulation must be replaced later by a more abstract
+one, for reasons which will become evident at a later stage.
+
+Since the introduction of the special principle of relativity has been
+justified, every intellect which strives after generalisation must feel
+the temptation to venture the step towards the general principle of
+relativity. But a simple and apparently quite reliable consideration
+seems to suggest that, for the present at any rate, there is little
+hope of success in such an attempt; Let us imagine ourselves
+transferred to our old friend the railway carriage, which is travelling
+at a uniform rate. As long as it is moving uniformly, the occupant of
+the carriage is not sensible of its motion, and it is for this reason
+that he can without reluctance interpret the facts of the case as
+indicating that the carriage is at rest, but the embankment in motion.
+Moreover, according to the special principle of relativity, this
+interpretation is quite justified also from a physical point of view.
+If the motion of the carriage is now changed into a non-uniform motion,
+as for instance by a powerful application of the brakes, then the
+occupant of the carriage experiences a correspondingly powerful jerk
+forwards. The retarded motion is manifested in the mechanical behaviour
+of bodies relative to the person in the railway carriage. The
+mechanical behaviour is different from that of the case previously
+considered, and for this reason it would appear to be impossible that
+the same mechanical laws hold relatively to the non-uniformly moving
+carriage, as hold with reference to the carriage when at rest or in
+uniform motion. At all events it is clear that the Galileian law does
+not hold with respect to the non-uniformly moving carriage. Because of
+this, we feel compelled at the present juncture to grant a kind of
+absolute physical reality to non-uniform motion, in opposition to the
+general principle of relativity. But in what follows we shall soon see
+that this conclusion cannot be maintained.
+
+
+XIX.
+
+THE GRAVITATIONAL FIELD
+
+“If we pick up a stone and then let it go, why does it fall to the
+ground?” The usual answer to this question is: “Because it is attracted
+by the earth.” Modern physics formulates the answer rather differently
+for the following reason. As a result of the more careful study of
+electromagnetic phenomena, we have come to regard action at a distance
+as a process impossible without the intervention of some intermediary
+medium. If, for instance, a magnet attracts a piece of iron, we cannot
+be content to regard this as meaning that the magnet acts directly on
+the iron through the intermediate empty space, but we are constrained
+to imagine—after the manner of Faraday—that the magnet always calls
+into being something physically real in the space around it, that
+something being what we call a “magnetic field.” In its turn this
+magnetic field operates on the piece of iron, so that the latter
+strives to move towards the magnet. We shall not discuss here the
+justification for this incidental conception, which is indeed a
+somewhat arbitrary one. We shall only mention that with its aid
+electromagnetic phenomena can be theoretically represented much more
+satisfactorily than without it, and this applies particularly to the
+transmission of electromagnetic waves. The effects of gravitation also
+are regarded in an analogous manner.
+
+The action of the earth on the stone takes place indirectly. The earth
+produces in its surrounding a gravitational field, which acts on the
+stone and produces its motion of fall. As we know from experience, the
+intensity of the action on a body dimishes according to a quite
+definite law, as we proceed farther and farther away from the earth.
+From our point of view this means: The law governing the properties of
+the gravitational field in space must be a perfectly definite one, in
+order correctly to represent the diminution of gravitational action
+with the distance from operative bodies. It is something like this: The
+body (_e.g._ the earth) produces a field in its immediate neighbourhood
+directly; the intensity and direction of the field at points farther
+removed from the body are thence determined by the law which governs
+the properties in space of the gravitational fields themselves.
+
+In contrast to electric and magnetic fields, the gravitational field
+exhibits a most remarkable property, which is of fundamental importance
+for what follows. Bodies which are moving under the sole influence of a
+gravitational field receive an acceleration, _which does not in the
+least depend either on the material or on the physical state of the
+body._ For instance, a piece of lead and a piece of wood fall in
+exactly the same manner in a gravitational field (_in vacuo_), when
+they start off from rest or with the same initial velocity. This law,
+which holds most accurately, can be expressed in a different form in
+the light of the following consideration.
+
+According to Newton’s law of motion, we have
+
+(Force) = (inertial mass) x (acceleration),
+
+where the “inertial mass” is a characteristic constant of the
+accelerated body. If now gravitation is the cause of the acceleration,
+we then have
+
+(Force) = (gravitational mass) x (intensity of the gravitational
+field),
+
+where the “gravitational mass” is likewise a characteristic constant
+for the body. From these two relations follows:
+
+image032
+
+
+If now, as we find from experience, the acceleration is to be
+independent of the nature and the condition of the body and always the
+same for a given gravitational field, then the ratio of the
+gravitational to the inertial mass must likewise be the same for all
+bodies. By a suitable choice of units we can thus make this ratio equal
+to unity. We then have the following law: The _gravitational_ mass of a
+body is equal to its _inertial_ mass.
+
+It is true that this important law had hitherto been recorded in
+mechanics, but it had not been _interpreted_. A satisfactory
+interpretation can be obtained only if we recognise the following fact:
+_The same_ quality of a body manifests itself according to
+circumstances as “inertia” or as “weight” (lit. “heaviness”). In the
+following section we shall show to what extent this is actually the
+case, and how this question is connected with the general postulate of
+relativity.
+
+
+XX.
+
+THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE
+GENERAL POSTULATE OF RELATIVITY
+
+We imagine a large portion of empty space, so far removed from stars
+and other appreciable masses, that we have before us approximately the
+conditions required by the fundamental law of Galilei. It is then
+possible to choose a Galileian reference-body for this part of space
+(world), relative to which points at rest remain at rest and points in
+motion continue permanently in uniform rectilinear motion. As
+reference-body let us imagine a spacious chest resembling a room with
+an observer inside who is equipped with apparatus. Gravitation
+naturally does not exist for this observer. He must fasten himself with
+strings to the floor, otherwise the slightest impact against the floor
+will cause him to rise slowly towards the ceiling of the room.
+
+To the middle of the lid of the chest is fixed externally a hook with
+rope attached, and now a “being” (what kind of a being is immaterial to
+us) begins pulling at this with a constant force. The chest together
+with the observer then begin to move “upwards” with a uniformly
+accelerated motion. In course of time their velocity will reach
+unheard-of values—provided that we are viewing all this from another
+reference-body which is not being pulled with a rope.
+
+But how does the man in the chest regard the Process? The acceleration
+of the chest will be transmitted to him by the reaction of the floor of
+the chest. He must therefore take up this pressure by means of his legs
+if he does not wish to be laid out full length on the floor. He is then
+standing in the chest in exactly the same way as anyone stands in a
+room of a home on our earth. If he releases a body which he previously
+had in his land, the accelertion of the chest will no longer be
+transmitted to this body, and for this reason the body will approach
+the floor of the chest with an accelerated relative motion. The
+observer will further convince himself _that the acceleration of the
+body towards the floor of the chest is always of the same magnitude,
+whatever kind of body he may happen to use for the experiment._
+
+Relying on his knowledge of the gravitational field (as it was
+discussed in the preceding section), the man in the chest will thus
+come to the conclusion that he and the chest are in a gravitational
+field which is constant with regard to time. Of course he will be
+puzzled for a moment as to why the chest does not fall in this
+gravitational field. just then, however, he discovers the hook in the
+middle of the lid of the chest and the rope which is attached to it,
+and he consequently comes to the conclusion that the chest is suspended
+at rest in the gravitational field.
+
+Ought we to smile at the man and say that he errs in his conclusion? I
+do not believe we ought to if we wish to remain consistent; we must
+rather admit that his mode of grasping the situation violates neither
+reason nor known mechanical laws. Even though it is being accelerated
+with respect to the “Galileian space” first considered, we can
+nevertheless regard the chest as being at rest. We have thus good
+grounds for extending the principle of relativity to include bodies of
+reference which are accelerated with respect to each other, and as a
+result we have gained a powerful argument for a generalised postulate
+of relativity.
+
+We must note carefully that the possibility of this mode of
+interpretation rests on the fundamental property of the gravitational
+field of giving all bodies the same acceleration, or, what comes to the
+same thing, on the law of the equality of inertial and gravitational
+mass. If this natural law did not exist, the man in the accelerated
+chest would not be able to interpret the behaviour of the bodies around
+him on the supposition of a gravitational field, and he would not be
+justified on the grounds of experience in supposing his reference-body
+to be “at rest.”
+
+Suppose that the man in the chest fixes a rope to the inner side of the
+lid, and that he attaches a body to the free end of the rope. The
+result of this will be to stretch the rope so that it will hang
+“vertically” downwards. If we ask for an opinion of the cause of
+tension in the rope, the man in the chest will say: “The suspended body
+experiences a downward force in the gravitational field, and this is
+neutralised by the tension of the rope; what determines the magnitude
+of the tension of the rope is the _gravitational mass_ of the suspended
+body.” On the other hand, an observer who is poised freely in space
+will interpret the condition of things thus: “The rope must perforce
+take part in the accelerated motion of the chest, and it transmits this
+motion to the body attached to it. The tension of the rope is just
+large enough to effect the acceleration of the body. That which
+determines the magnitude of the tension of the rope is the _inertial
+mass_ of the body.” Guided by this example, we see that our extension
+of the principle of relativity implies the _necessity_ of the law of
+the equality of inertial and gravitational mass. Thus we have obtained
+a physical interpretation of this law.
+
+From our consideration of the accelerated chest we see that a general
+theory of relativity must yield important results on the laws of
+gravitation. In point of fact, the systematic pursuit of the general
+idea of relativity has supplied the laws satisfied by the gravitational
+field. Before proceeding farther, however, I must warn the reader
+against a misconception suggested by these considerations. A
+gravitational field exists for the man in the chest, despite the fact
+that there was no such field for the co-ordinate system first chosen.
+Now we might easily suppose that the existence of a gravitational field
+is always only an _apparent_ one. We might also think that, regardless
+of the kind of gravitational field which may be present, we could
+always choose another reference-body such that _no_ gravitational field
+exists with reference to it. This is by no means true for all
+gravitational fields, but only for those of quite special form. It is,
+for instance, impossible to choose a body of reference such that, as
+judged from it, the gravitational field of the earth (in its entirety)
+vanishes.
+
+We can now appreciate why that argument is not convincing, which we
+brought forward against the general principle of relativity at the end
+of Section XVIII. It is certainly true that the observer in the railway
+carriage experiences a jerk forwards as a result of the application of
+the brake, and that he recognises, in this the non-uniformity of motion
+(retardation) of the carriage. But he is compelled by nobody to refer
+this jerk to a “real” acceleration (retardation) of the carriage. He
+might also interpret his experience thus: “My body of reference (the
+carriage) remains permanently at rest. With reference to it, however,
+there exists (during the period of application of the brakes) a
+gravitational field which is directed forwards and which is variable
+with respect to time. Under the influence of this field, the embankment
+together with the earth moves non-uniformly in such a manner that their
+original velocity in the backwards direction is continuously reduced.”
+
+
+XXI.
+
+IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
+SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
+
+We have already stated several times that classical mechanics starts
+out from the following law: Material particles sufficiently far removed
+from other material particles continue to move uniformly in a straight
+line or continue in a state of rest. We have also repeatedly emphasised
+that this fundamental law can only be valid for bodies of reference _K_
+which possess certain unique states of motion, and which are in uniform
+translational motion relative to each other. Relative to other
+reference-bodies _K_ the law is not valid. Both in classical mechanics
+and in the special theory of relativity we therefore differentiate
+between reference-bodies _K_ relative to which the recognised “laws of
+nature” can be said to hold, and reference-bodies _K_ relative to which
+these laws do not hold.
+
+But no person whose mode of thought is logical can rest satisfied with
+this condition of things. He asks: “How does it come that certain
+reference-bodies (or their states of motion) are given priority over
+other reference-bodies (or their states of motion)? _What is the reason
+for this preference?_” In order to show clearly what I mean by this
+question, I shall make use of a comparison.
+
+I am standing in front of a gas range. Standing alongside of each other
+on the range are two pans so much alike that one may be mistaken for
+the other. Both are half full of water. I notice that steam is being
+emitted continuously from the one pan, but not from the other. I am
+surprised at this, even if I have never seen either a gas range or a
+pan before. But if I now notice a luminous something of bluish colour
+under the first pan but not under the other, I cease to be astonished,
+even if I have never before seen a gas flame. For I can only say that
+this bluish something will cause the emission of the steam, or at least
+_possibly_ it may do so. If, however, I notice the bluish something in
+neither case, and if I observe that the one continuously emits steam
+whilst the other does not, then I shall remain astonished and
+dissatisfied until I have discovered some circumstance to which I can
+attribute the different behaviour of the two pans.
+
+Analogously, I seek in vain for a real something in classical mechanics
+(or in the special theory of relativity) to which I can attribute the
+different behaviour of bodies considered with respect to the reference
+systems _K_ and _K′_.[15] Newton saw this objection and attempted to
+invalidate it, but without success. But E. Mach recognised it most
+clearly of all, and because of this objection he claimed that mechanics
+must be placed on a new basis. It can only be got rid of by means of a
+physics which is conformable to the general principle of relativity,
+since the equations of such a theory hold for every body of reference,
+whatever may be its state of motion.
+
+
+ [15] The objection is of importance more especially when the state of
+ motion of the reference-body is of such a nature that it does not
+ require any external agency for its maintenance, _e.g._ in the case
+ when the reference-body is rotating uniformly.
+
+
+XXII.
+
+A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
+
+The considerations of Section XX show that the general principle of
+relativity puts us in a position to derive properties of the
+gravitational field in a purely theoretical manner. Let us suppose, for
+instance, that we know the space-time “course” for any natural process
+whatsoever, as regards the manner in which it takes place in the
+Galileian domain relative to a Galileian body of reference _K_. By
+means of purely theoretical operations (_i.e._ simply by calculation)
+we are then able to find how this known natural process appears, as
+seen from a reference-body _K′_ which is accelerated relatively to _K_.
+But since a gravitational field exists with respect to this new body of
+reference _K′_, our consideration also teaches us how the gravitational
+field influences the process studied.
+
+For example, we learn that a body which is in a state of uniform
+rectilinear motion with respect to _K_ (in accordance with the law of
+Galilei) is executing an accelerated and in general curvilinear motion
+with respect to the accelerated reference-body _K′_ (chest). This
+acceleration or curvature corresponds to the influence on the moving
+body of the gravitational field prevailing relatively to _K_. It is
+known that a gravitational field influences the movement of bodies in
+this way, so that our consideration supplies us with nothing
+essentially new.
+
+However, we obtain a new result of fundamental importance when we carry
+out the analogous consideration for a ray of light. With respect to the
+Galileian reference-body _K_, such a ray of light is transmitted
+rectilinearly with the velocity _c_. It can easily be shown that the
+path of the same ray of light is no longer a straight line when we
+consider it with reference to the accelerated chest (reference-body
+_K′_). From this we conclude, _that, in general, rays of light are
+propagated curvilinearly in gravitational fields._ In two respects this
+result is of great importance.
+
+In the first place, it can be compared with the reality. Although a
+detailed examination of the question shows that the curvature of light
+rays required by the general theory of relativity is only exceedingly
+small for the gravitational fields at our disposal in practice, its
+estimated magnitude for light rays passing the sun at grazing incidence
+is nevertheless 1.7 seconds of arc. This ought to manifest itself in
+the following way. As seen from the earth, certain fixed stars appear
+to be in the neighbourhood of the sun, and are thus capable of
+observation during a total eclipse of the sun. At such times, these
+stars ought to appear to be displaced outwards from the sun by an
+amount indicated above, as compared with their apparent position in the
+sky when the sun is situated at another part of the heavens. The
+examination of the correctness or otherwise of this deduction is a
+problem of the greatest importance, the early solution of which is to
+be expected of astronomers.[16]
+
+
+ [16] By means of the star photographs of two expeditions equipped by a
+ Joint Committee of the Royal and Royal Astronomical Societies, the
+ existence of the deflection of light demanded by theory was first
+ confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
+ III.)
+
+
+In the second place our result shows that, according to the general
+theory of relativity, the law of the constancy of the velocity of light
+in vacuo, which constitutes one of the two fundamental assumptions in
+the special theory of relativity and to which we have already
+frequently referred, cannot claim any unlimited validity. A curvature
+of rays of light can only take place when the velocity of propagation
+of light varies with position. Now we might think that as a consequence
+of this, the special theory of relativity and with it the whole theory
+of relativity would be laid in the dust. But in reality this is not the
+case. We can only conclude that the special theory of relativity cannot
+claim an unlimited domain of validity; its results hold only so long as
+we are able to disregard the influences of gravitational fields on the
+phenomena (_e.g._ of light).
+
+Since it has often been contended by opponents of the theory of
+relativity that the special theory of relativity is overthrown by the
+general theory of relativity, it is perhaps advisable to make the facts
+of the case clearer by means of an appropriate comparison. Before the
+development of electrodynamics the laws of electrostatics were looked
+upon as the laws of electricity. At the present time we know that
+electric fields can be derived correctly from electrostatic
+considerations only for the case, which is never strictly realised, in
+which the electrical masses are quite at rest relatively to each other,
+and to the co-ordinate system. Should we be justified in saying that
+for this reason electrostatics is overthrown by the field-equations of
+Maxwell in electrodynamics? Not in the least. Electrostatics is
+contained in electrodynamics as a limiting case; the laws of the latter
+lead directly to those of the former for the case in which the fields
+are invariable with regard to time. No fairer destiny could be allotted
+to any physical theory, than that it should of itself point out the way
+to the introduction of a more comprehensive theory, in which it lives
+on as a limiting case.
+
+In the example of the transmission of light just dealt with, we have
+seen that the general theory of relativity enables us to derive
+theoretically the influence of a gravitational field on the course of
+natural processes, the laws of which are already known when a
+gravitational field is absent. But the most attractive problem, to the
+solution of which the general theory of relativity supplies the key,
+concerns the investigation of the laws satisfied by the gravitational
+field itself. Let us consider this for a moment.
+
+We are acquainted with space-time domains which behave (approximately)
+in a “Galileian” fashion under suitable choice of reference-body,
+_i.e._ domains in which gravitational fields are absent. If we now
+refer such a domain to a reference-body _K′_ possessing any kind of
+motion, then relative to _K′_ there exists a gravitational field which
+is variable with respect to space and time.[17] The character of this
+field will of course depend on the motion chosen for _K′._ According to
+the general theory of relativity, the general law of the gravitational
+field must be satisfied for all gravitational fields obtainable in this
+way. Even though by no means all gravitationial fields can be produced
+in this way, yet we may entertain the hope that the general law of
+gravitation will be derivable from such gravitational fields of a
+special kind. This hope has been realised in the most beautiful manner.
+But between the clear vision of this goal and its actual realisation it
+was necessary to surmount a serious difficulty, and as this lies deep
+at the root of things, I dare not withhold it from the reader. We
+require to extend our ideas of the space-time continuum still farther.
+
+
+ [17] This follows from a generalisation of the discussion in Section
+ XX.
+
+
+XXIII.
+
+BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
+
+Hitherto I have purposely refrained from speaking about the physical
+interpretation of space- and time-data in the case of the general
+theory of relativity. As a consequence, I am guilty of a certain
+slovenliness of treatment, which, as we know from the special theory of
+relativity, is far from being unimportant and pardonable. It is now
+high time that we remedy this defect; but I would mention at the
+outset, that this matter lays no small claims on the patience and on
+the power of abstraction of the reader.
+
+We start off again from quite special cases, which we have frequently
+used before. Let us consider a space time domain in which no
+gravitational field exists relative to a reference-body _K_ whose state
+of motion has been suitably chosen. _K_ is then a Galileian
+reference-body as regards the domain considered, and the results of the
+special theory of relativity hold relative to _K_. Let us suppose the
+same domain referred to a second body of reference _K′_, which is
+rotating uniformly with respect to _K_. In order to fix our ideas, we
+shall imagine _K′_ to be in the form of a plane circular disc, which
+rotates uniformly in its own plane about its centre. An observer who is
+sitting eccentrically on the disc _K′_ is sensible of a force which
+acts outwards in a radial direction, and which would be interpreted as
+an effect of inertia (centrifugal force) by an observer who was at rest
+with respect to the original reference-body _K_. But the observer on
+the disc may regard his disc as a reference-body which is “at rest”; on
+the basis of the general principle of relativity he is justified in
+doing this. The force acting on himself, and in fact on all other
+bodies which are at rest relative to the disc, he regards as the effect
+of a gravitational field. Nevertheless, the space-distribution of this
+gravitational field is of a kind that would not be possible on Newton’s
+theory of gravitation.[18] But since the observer believes in the
+general theory of relativity, this does not disturb him; he is quite in
+the right when he believes that a general law of gravitation can be
+formulated—a law which not only explains the motion of the stars
+correctly, but also the field of force experienced by himself.
+
+
+ [18] The field disappears at the centre of the disc and increases
+ proportionally to the distance from the centre as we proceed outwards.
+
+
+The observer performs experiments on his circular disc with clocks and
+measuring-rods. In doing so, it is his intention to arrive at exact
+definitions for the signification of time- and space-data with
+reference to the circular disc _K′_, these definitions being based on
+his observations. What will be his experience in this enterprise?
+
+To start with, he places one of two identically constructed clocks at
+the centre of the circular disc, and the other on the edge of the disc,
+so that they are at rest relative to it. We now ask ourselves whether
+both clocks go at the same rate from the standpoint of the non-rotating
+Galileian reference-body _K_. As judged from this body, the clock at
+the centre of the disc has no velocity, whereas the clock at the edge
+of the disc is in motion relative to _K_ in consequence of the
+rotation. According to a result obtained in Section XII, it follows
+that the latter clock goes at a rate permanently slower than that of
+the clock at the centre of the circular disc, _i.e._ as observed from
+_K_. It is obvious that the same effect would be noted by an observer
+whom we will imagine sitting alongside his clock at the centre of the
+circular disc. Thus on our circular disc, or, to make the case more
+general, in every gravitational field, a clock will go more quickly or
+less quickly, according to the position in which the clock is situated
+(at rest). For this reason it is not possible to obtain a reasonable
+definition of time with the aid of clocks which are arranged at rest
+with respect to the body of reference. A similar difficulty presents
+itself when we attempt to apply our earlier definition of simultaneity
+in such a case, but I do not wish to go any farther into this question.
+
+Moreover, at this stage the definition of the space co-ordinates also
+presents insurmountable difficulties. If the observer applies his
+standard measuring-rod (a rod which is short as compared with the
+radius of the disc) tangentially to the edge of the disc, then, as
+judged from the Galileian system, the length of this rod will be less
+than 1, since, according to Section XII, moving bodies suffer a
+shortening in the direction of the motion. On the other hand, the
+measuring-rod will not experience a shortening in length, as judged
+from _K_, if it is applied to the disc in the direction of the radius.
+If, then, the observer first measures the circumference of the disc
+with his measuring-rod and then the diameter of the disc, on dividing
+the one by the other, he will not obtain as quotient the familiar
+number π = 3.14 . . ., but a larger number,[19] whereas of course, for
+a disc which is at rest with respect to _K_, this operation would yield
+π exactly. This proves that the propositions of Euclidean geometry
+cannot hold exactly on the rotating disc, nor in general in a
+gravitational field, at least if we attribute the length 1 to the rod
+in all positions and in every orientation. Hence the idea of a straight
+line also loses its meaning. We are therefore not in a position to
+define exactly the co-ordinates _x, y, z_ relative to the disc by means
+of the method used in discussing the special theory, and as long as the
+co-ordinates and times of events have not been defined, we cannot
+assign an exact meaning to the natural laws in which these occur.
+
+
+ [19] Throughout this consideration we have to use the Galileian
+ (non-rotating) system _K_ as reference-body, since we may only assume
+ the validity of the results of the special theory of relativity
+ relative to _K_ (relative to _K′_ a gravitational field prevails).
+
+
+Thus all our previous conclusions based on general relativity would
+appear to be called in question. In reality we must make a subtle
+detour in order to be able to apply the postulate of general relativity
+exactly. I shall prepare the reader for this in the following
+paragraphs.
+
+
+XXIV.
+
+EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
+
+The surface of a marble table is spread out in front of me. I can get
+from any one point on this table to any other point by passing
+continuously from one point to a “neighbouring” one, and repeating this
+process a (large) number of times, or, in other words, by going from
+point to point without executing “jumps.” I am sure the reader will
+appreciate with sufficient clearness what I mean here by “neighbouring”
+and by “jumps” (if he is not too pedantic). We express this property of
+the surface by describing the latter as a continuum.
+
+Let us now imagine that a large number of little rods of equal length
+have been made, their lengths being small compared with the dimensions
+of the marble slab. When I say they are of equal length, I mean that
+one can be laid on any other without the ends overlapping. We next lay
+four of these little rods on the marble slab so that they constitute a
+quadrilateral figure (a square), the diagonals of which are equally
+long. To ensure the equality of the diagonals, we make use of a little
+testing-rod. To this square we add similar ones, each of which has one
+rod in common with the first. We proceed in like manner with each of
+these squares until finally the whole marble slab is laid out with
+squares. The arrangement is such, that each side of a square belongs to
+two squares and each corner to four squares.
+
+It is a veritable wonder that we can carry out this business without
+getting into the greatest difficulties. We only need to think of the
+following. If at any moment three squares meet at a corner, then two
+sides of the fourth square are already laid, and, as a consequence, the
+arrangement of the remaining two sides of the square is already
+completely determined. But I am now no longer able to adjust the
+quadrilateral so that its diagonals may be equal. If they are equal of
+their own accord, then this is an especial favour of the marble slab
+and of the little rods, about which I can only be thankfully surprised.
+We must experience many such surprises if the construction is to be
+successful.
+
+If everything has really gone smoothly, then I say that the points of
+the marble slab constitute a Euclidean continuum with respect to the
+little rod, which has been used as a “distance” (line-interval). By
+choosing one corner of a square as “origin” I can characterise every
+other corner of a square with reference to this origin by means of two
+numbers. I only need state how many rods I must pass over when,
+starting from the origin, I proceed towards the “right” and then
+“upwards,” in order to arrive at the corner of the square under
+consideration. These two numbers are then the “Cartesian co-ordinates”
+of this corner with reference to the “Cartesian co-ordinate system”
+which is determined by the arrangement of little rods.
+
+By making use of the following modification of this abstract
+experiment, we recognise that there must also be cases in which the
+experiment would be unsuccessful. We shall suppose that the rods
+“expand” by in amount proportional to the increase of temperature. We
+heat the central part of the marble slab, but not the periphery, in
+which case two of our little rods can still be brought into coincidence
+at every position on the table. But our construction of squares must
+necessarily come into disorder during the heating, because the little
+rods on the central region of the table expand, whereas those on the
+outer part do not.
+
+With reference to our little rods—defined as unit lengths—the marble
+slab is no longer a Euclidean continuum, and we are also no longer in
+the position of defining Cartesian co-ordinates directly with their
+aid, since the above construction can no longer be carried out. But
+since there are other things which are not influenced in a similar
+manner to the little rods (or perhaps not at all) by the temperature of
+the table, it is possible quite naturally to maintain the point of view
+that the marble slab is a “Euclidean continuum.” This can be done in a
+satisfactory manner by making a more subtle stipulation about the
+measurement or the comparison of lengths.
+
+But if rods of every kind (_i.e._ of every material) were to behave _in
+the same way_ as regards the influence of temperature when they are on
+the variably heated marble slab, and if we had no other means of
+detecting the effect of temperature than the geometrical behaviour of
+our rods in experiments analogous to the one described above, then our
+best plan would be to assign the distance one to two points on the
+slab, provided that the ends of one of our rods could be made to
+coincide with these two points; for how else should we define the
+distance without our proceeding being in the highest measure grossly
+arbitrary? The method of Cartesian coordinates must then be discarded,
+and replaced by another which does not assume the validity of Euclidean
+geometry for rigid bodies.[20] The reader will notice that the
+situation depicted here corresponds to the one brought about by the
+general postulate of relativity (Section XXIII).
+
+
+ [20] Mathematicians have been confronted with our problem in the
+ following form. If we are given a surface (_e.g._ an ellipsoid) in
+ Euclidean three-dimensional space, then there exists for this surface
+ a two-dimensional geometry, just as much as for a plane surface. Gauss
+ undertook the task of treating this two-dimensional geometry from
+ first principles, without making use of the fact that the surface
+ belongs to a Euclidean continuum of three dimensions. If we imagine
+ constructions to be made with rigid rods _in the surface_ (similar to
+ that above with the marble slab), we should find that different laws
+ hold for these from those resulting on the basis of Euclidean plane
+ geometry. The surface is not a Euclidean continuum with respect to the
+ rods, and we cannot define Cartesian co-ordinates _in the surface_.
+ Gauss indicated the principles according to which we can treat the
+ geometrical relationships in the surface, and thus pointed out the way
+ to the method of Riemann of treating multi-dimensional, non-Euclidean
+ _continuum_. Thus it is that mathematicians long ago solved the formal
+ problems to which we are led by the general postulate of relativity.
+
+
+XXV.
+
+GAUSSIAN CO-ORDINATES
+
+image033
+
+
+According to Gauss, this combined analytical and geometrical mode of
+handling the problem can be arrived at in the following way. We imagine
+a system of arbitrary curves (see Fig. 4) drawn on the surface of the
+table. These we designate as _u_-curves, and we indicate each of them
+by means of a number. The Curves _u_ = 1, _u_ = 2 and _u_ = 3 are drawn
+in the diagram. Between the curves _u_ = 1 and _u_ = 2 we must imagine
+an infinitely large number to be drawn, all of which correspond to real
+numbers lying between 1 and 2. We have then a system of _u_-curves, and
+this “infinitely dense” system covers the whole surface of the table.
+These _u_-curves must not intersect each other, and through each point
+of the surface one and only one curve must pass. Thus a perfectly
+definite value of _u_ belongs to every point on the surface of the
+marble slab. In like manner we imagine a system of _v_-curves drawn on
+the surface. These satisfy the same conditions as the _u_-curves, they
+are provided with numbers in a corresponding manner, and they may
+likewise be of arbitrary shape. It follows that a value of _u_ and a
+value of _v_ belong to every point on the surface of the table. We call
+these two numbers the co-ordinates of the surface of the table
+(Gaussian co-ordinates). For example, the point _P_ in the diagram has
+the Gaussian co-ordinates _u_ = 3, _v_ = 1. Two neighbouring points _P_
+and _P′_ on the surface then correspond to the co-ordinates
+
+_P_: _u, v_
+
+_P′_: _u_ + _du, v_ + _dv_,
+
+where _du_ and _dv_ signify very small numbers. In a similar manner we
+may indicate the distance (line-interval) between _P_ and _P′_, as
+measured with a little rod, by means of the very small number _ds_.
+Then according to Gauss we have
+
+_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2,
+
+where _g_11, _g_12, _g_22, are magnitudes which depend in a perfectly
+definite way on _u_ and _v_. The magnitudes _g_11, _g_12 and _g_22,
+determine the behaviour of the rods relative to the _u_-curves and
+_v_-curves, and thus also relative to the surface of the table. For the
+case in which the points of the surface considered form a Euclidean
+continuum with reference to the measuring-rods, but only in this case,
+it is possible to draw the _u_-curves and _v_-curves and to attach
+numbers to them, in such a manner, that we simply have:
+
+_ds_2 = _du_2 + _dv_2
+
+Under these conditions, the _u_-curves and _v_-curves are straight
+lines in the sense of Euclidean geometry, and they are perpendicular to
+each other. Here the Gaussian coordinates are simply Cartesian ones. It
+is clear that Gauss co-ordinates are nothing more than an association
+of two sets of numbers with the points of the surface considered, of
+such a nature that numerical values differing very slightly from each
+other are associated with neighbouring points “in space.”
+
+So far, these considerations hold for a continuum of two dimensions.
+But the Gaussian method can be applied also to a continuum of three,
+four or more dimensions. If, for instance, a continuum of four
+dimensions be supposed available, we may represent it in the following
+way. With every point of the continuum, we associate arbitrarily four
+numbers, _x_1, _x_2, _x_3, _x_4, which are known as “co-ordinates.”
+Adjacent points correspond to adjacent values of the coordinates. If a
+distance _ds_ is associated with the adjacent points _P_ and _P′_, this
+distance being measurable and well defined from a physical point of
+view, then the following formula holds:
+
+_ds_2 = _g_11_dx_12 + 2_g_12_dx_1_dx_2 . . . . + _g_44_dx_42,
+
+where the magnitudes _g_11, etc., have values which vary with the
+position in the continuum. Only when the continuum is a Euclidean one
+is it possible to associate the co-ordinates _x_1 . . _x_4. with the
+points of the continuum so that we have simply
+
+_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
+
+In this case relations hold in the four-dimensional continuum which are
+analogous to those holding in our three-dimensional measurements.
+
+However, the Gauss treatment for _ds_2 which we have given above is not
+always possible. It is only possible when sufficiently small regions of
+the continuum under consideration may be regarded as Euclidean
+continua. For example, this obviously holds in the case of the marble
+slab of the table and local variation of temperature. The temperature
+is practically constant for a small part of the slab, and thus the
+geometrical behaviour of the rods is _almost_ as it ought to be
+according to the rules of Euclidean geometry. Hence the imperfections
+of the construction of squares in the previous section do not show
+themselves clearly until this construction is extended over a
+considerable portion of the surface of the table.
+
+We can sum this up as follows: Gauss invented a method for the
+mathematical treatment of continua in general, in which
+“size-relations” (“distances” between neighbouring points) are defined.
+To every point of a continuum are assigned as many numbers (Gaussian
+coordinates) as the continuum has dimensions. This is done in such a
+way, that only one meaning can be attached to the assignment, and that
+numbers (Gaussian coordinates) which differ by an indefinitely small
+amount are assigned to adjacent points. The Gaussian coordinate system
+is a logical generalisation of the Cartesian co-ordinate system. It is
+also applicable to non-Euclidean continua, but only when, with respect
+to the defined “size” or “distance,” small parts of the continuum under
+consideration behave more nearly like a Euclidean system, the smaller
+the part of the continuum under our notice.
+
+
+XXVI.
+
+THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED
+AS A EUCLIDEAN CONTINUUM
+
+We are now in a position to formulate more exactly the idea of
+Minkowski, which was only vaguely indicated in Section XVII. In
+accordance with the special theory of relativity, certain co-ordinate
+systems are given preference for the description of the
+four-dimensional, space-time continuum. We called these “Galileian
+co-ordinate systems.” For these systems, the four co-ordinates _x, y,
+z, t_, which determine an event or—in other words—a point of the
+four-dimensional continuum, are defined physically in a simple manner,
+as set forth in detail in the first part of this book. For the
+transition from one Galileian system to another, which is moving
+uniformly with reference to the first, the equations of the Lorentz
+transformation are valid. These last form the basis for the derivation
+of deductions from the special theory of relativity, and in themselves
+they are nothing more than the expression of the universal validity of
+the law of transmission of light for all Galileian systems of
+reference.
+
+Minkowski found that the Lorentz transformations satisfy the following
+simple conditions. Let us consider two neighbouring events, the
+relative position of which in the four-dimensional continuum is given
+with respect to a Galileian reference-body _K_ by the space co-ordinate
+differences _dx, dy, dz_ and the time-difference _dt_. With reference
+to a second Galileian system we shall suppose that the corresponding
+differences for these two events are _dx′, dy′, dz′, dt′_. Then these
+magnitudes always fulfill the condition.[21]
+
+
+ [21] Cf. Appendixes I and II. The relations which are derived there
+ for the co-ordinates themselves are valid also for co-ordinate
+ _differences_, and thus also for co-ordinate differentials
+ (indefinitely small differences).
+
+
+_dx_2 + _dy_2 + _dz_2 – _c_2_dt_2 = _dx′_2 + _dy′_2 + _dz′_2 –
+_c_2_dt′_2.
+
+
+The validity of the Lorentz transformation follows from this condition.
+We can express this as follows: The magnitude
+
+_ds_2 = _dx_2 + _dy_2 + _dz_2 – _c_2 _dt_2,
+
+
+which belongs to two adjacent points of the four-dimensional space-time
+continuum, has the same value for all selected (Galileian)
+reference-bodies. If we replace _x, y, z_,
+
+image034
+
+
+by _x_1, _x_2, _x_3, _x_4, we also obtain the result that
+
+_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
+
+
+is independent of the choice of the body of reference. We call the
+magnitude _ds_ the “distance” apart of the two events or
+four-dimensional points.
+
+Thus, if we choose as time-variable the imaginary variable
+
+image035
+
+
+instead of the real quantity _t_, we can regard the space-time
+contintium—accordance with the special theory of relativity—as a
+“Euclidean” four-dimensional continuum, a result which follows from the
+considerations of the preceding section.
+
+
+XXVII.
+
+THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A
+EUCLIDEAN CONTINUUM
+
+In the first part of this book we were able to make use of space-time
+co-ordinates which allowed of a simple and direct physical
+interpretation, and which, according to Section XXVI, can be regarded
+as four-dimensional Cartesian co-ordinates. This was possible on the
+basis of the law of the constancy of the velocity of light. But
+according to Section XXI the general theory of relativity cannot retain
+this law. On the contrary, we arrived at the result that according to
+this latter theory the velocity of light must always depend on the
+co-ordinates when a gravitational field is present. In connection with
+a specific illustration in Section XXIII, we found that the presence of
+a gravitational field invalidates the definition of the coordinates and
+the time, which led us to our objective in the special theory of
+relativity.
+
+In view of the resuIts of these considerations we are led to the
+conviction that, according to the general principle of relativity, the
+space-time continuum cannot be regarded as a Euclidean one, but that
+here we have the general case, corresponding to the marble slab with
+local variations of temperature, and with which we made acquaintance as
+an example of a two-dimensional continuum. Just as it was there
+impossible to construct a Cartesian co-ordinate system from equal rods,
+so here it is impossible to build up a system (reference-body) from
+rigid bodies and clocks, which shall be of such a nature that
+measuring-rods and clocks, arranged rigidly with respect to one
+another, shall indicate position and time directly. Such was the
+essence of the difficulty with which we were confronted in Section
+XXIII.
+
+But the considerations of Sections XXV and XXVI show us the way to
+surmount this difficulty. We refer the four-dimensional space-time
+continuum in an arbitrary manner to Gauss co-ordinates. We assign to
+every point of the continuum (event) four numbers, _x_1, _x_2, _x_3,
+_x_4 (co-ordinates), which have not the least direct physical
+significance, but only serve the purpose of numbering the points of the
+continuum in a definite but arbitrary manner. This arrangement does not
+even need to be of such a kind that we must regard _x_1, _x_2, _x_3, as
+“space” co-ordinates and _x_4, as a “time” co-ordinate.
+
+The reader may think that such a description of the world would be
+quite inadequate. What does it mean to assign to an event the
+particular co-ordinates _x_1, _x_2, _x_3, _x_4, if in themselves these
+co-ordinates have no significance? More careful consideration shows,
+however, that this anxiety is unfounded. Let us consider, for instance,
+a material point with any kind of motion. If this point had only a
+momentary existence without duration, then it would to described in
+space-time by a single system of values _x_1, _x_2, _x_3, _x_4. Thus
+its permanent existence must be characterised by an infinitely large
+number of such systems of values, the co-ordinate values of which are
+so close together as to give continuity; corresponding to the material
+point, we thus have a (uni-dimensional) line in the four-dimensional
+continuum. In the same way, any such lines in our continuum correspond
+to many points in motion. The only statements having regard to these
+points which can claim a physical existence are in reality the
+statements about their encounters. In our mathematical treatment, such
+an encounter is expressed in the fact that the two lines which
+represent the motions of the points in question have a particular
+system of co-ordinate values, _x_1, _x_2, _x_3, _x_4, in common. After
+mature consideration the reader will doubtless admit that in reality
+such encounters constitute the only actual evidence of a time-space
+nature with which we meet in physical statements.
+
+When we were describing the motion of a material point relative to a
+body of reference, we stated nothing more than the encounters of this
+point with particular points of the reference-body. We can also
+determine the corresponding values of the time by the observation of
+encounters of the body with clocks, in conjunction with the observation
+of the encounter of the hands of clocks with particular points on the
+dials. It is just the same in the case of space-measurements by means
+of measuring-rods, as a little consideration will show.
+
+The following statements hold generally: Every physical description
+resolves itself into a number of statements, each of which refers to
+the space-time coincidence of two events _A_ and _B_. In terms of
+Gaussian co-ordinates, every such statement is expressed by the
+agreement of their four co-ordinates _x_1, _x_2, _x_3, _x_4. Thus in
+reality, the description of the time-space continuum by means of Gauss
+co-ordinates completely replaces the description with the aid of a body
+of reference, without suffering from the defects of the latter mode of
+description; it is not tied down to the Euclidean character of the
+continuum which has to be represented.
+
+
+XXVIII.
+
+EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
+
+We are now in a position to replace the provisional formulation of the
+general principle of relativity given in Section XVIII by an exact
+formulation. The form there used, “All bodies of reference _K, K′_,
+etc., are equivalent for the description of natural phenomena
+(formulation of the general laws of nature), whatever may be their
+state of motion,” cannot be maintained, because the use of rigid
+reference-bodies, in the sense of the method followed in the special
+theory of relativity, is in general not possible in space-time
+description. The Gauss co-ordinate system has to take the place of the
+body of reference. The following statement corresponds to the
+fundamental idea of the general principle of relativity: “_All Gaussian
+co-ordinate systems are essentially equivalent for the formulation of
+the general laws of nature._”
+
+We can state this general principle of relativity in still another
+form, which renders it yet more clearly intelligible than it is when in
+the form of the natural extension of the special principle of
+relativity. According to the special theory of relativity, the
+equations which express the general laws of nature pass over into
+equations of the same form when, by making use of the Lorentz
+transformation, we replace the space-time variables _x, y, z, t_, of a
+(Galileian) reference-body _K_ by the space-time variables _x′, y′, z′,
+t′_, of a new reference-body _K′_. According to the general theory of
+relativity, on the other hand, by application of _arbitrary
+substitutions_ of the Gauss variables _x_1, _x_2, _x_3, _x_4, the
+equations must pass over into equations of the same form; for every
+transformation (not only the Lorentz transformation) corresponds to the
+transition of one Gauss co-ordinate system into another.
+
+If we desire to adhere to our “old-time” three-dimensional view of
+things, then we can characterise the development which is being
+undergone by the fundamental idea of the general theory of relativity
+as follows: The special theory of relativity has reference to Galileian
+domains, _i.e._ to those in which no gravitational field exists. In
+this connection a Galileian reference-body serves as body of reference,
+_i.e._ a rigid body the state of motion of which is so chosen that the
+Galileian law of the uniform rectilinear motion of “isolated” material
+points holds relatively to it.
+
+Certain considerations suggest that we should refer the same Galileian
+domains to _non-Galileian_ reference-bodies also. A gravitational field
+of a special kind is then present with respect to these bodies (cf.
+Sections XX and XXIII).
+
+In gravitational fields there are no such things as rigid bodies with
+Euclidean properties; thus the fictitious rigid body of reference is of
+no avail in the general theory of relativity. The motion of clocks is
+also influenced by gravitational fields, and in such a way that a
+physical definition of time which is made directly with the aid of
+clocks has by no means the same degree of plausibility as in the
+special theory of relativity.
+
+For this reason non-rigid reference-bodies are used, which are as a
+whole not only moving in any way whatsoever, but which also suffer
+alterations in form _ad lib._ during their motion. Clocks, for which
+the law of motion is of any kind, however irregular, serve for the
+definition of time. We have to imagine each of these clocks fixed at a
+point on the non-rigid reference-body. These clocks satisfy only the
+one condition, that the “readings” which are observed simultaneously on
+adjacent clocks (in space) differ from each other by an indefinitely
+small amount. This non-rigid reference-body, which might appropriately
+be termed a “reference-mollusc”, is in the main equivalent to a
+Gaussian four-dimensional co-ordinate system chosen arbitrarily. That
+which gives the “mollusc” a certain comprehensibility as compared with
+the Gauss co-ordinate system is the (really unjustified) formal
+retention of the separate existence of the
+
+space co-ordinates as opposed to the time co-ordinate. Every point on
+the mollusc is treated as a space-point, and every material point which
+is at rest relatively to it as at rest, so long as the mollusc is
+considered as reference-body. The general principle of relativity
+requires that all these molluscs can be used as reference-bodies with
+equal right and equal success in the formulation of the general laws of
+nature; the laws themselves must be quite independent of the choice of
+mollusc.
+
+The great power possessed by the general principle of relativity lies
+in the comprehensive limitation which is imposed on the laws of nature
+in consequence of what we have seen above.
+
+
+XXIX.
+
+THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
+PRINCIPLE OF RELATIVITY
+
+If the reader has followed all our previous considerations, he will
+have no further difficulty in understanding the methods leading to the
+solution of the problem of gravitation.
+
+We start off on a consideration of a Galileian domain, _i.e._ a domain
+in which there is no gravitational field relative to the Galileian
+reference-body _K_. The behaviour of measuring-rods and clocks with
+reference to _K_ is known from the special theory of relativity,
+likewise the behaviour of “isolated” material points; the latter move
+uniformly and in straight lines.
+
+Now let us refer this domain to a random Gauss coordinate system or to
+a “mollusc” as reference-body _K′_. Then with respect to _K′_ there is
+a gravitational field _G_ (of a particular kind). We learn the
+behaviour of measuring-rods and clocks and also of freely-moving
+material points with reference to _K′_ simply by mathematical
+transformation. We interpret this behaviour as the behaviour of
+measuring-rods, clocks and material points under the influence of the
+gravitational field _G_. Hereupon we introduce a hypothesis: that the
+influence of the gravitational field on measuring-rods, clocks and
+freely-moving material points continues to take place according to the
+same laws, even in the case where the prevailing gravitational field is
+_not_ derivable from the Galileian special case, simply by means of a
+transformation of co-ordinates.
+
+The next step is to investigate the space-time behaviour of the
+gravitational field _G_, which was derived from the Galileian special
+case simply by transformation of the coordinates. This behaviour is
+formulated in a law, which is always valid, no matter how the
+reference-body (mollusc) used in the description may be chosen.
+
+This law is not yet the _general_ law of the gravitational field, since
+the gravitational field under consideration is of a special kind. In
+order to find out the general law-of-field of gravitation we still
+require to obtain a generalisation of the law as found above. This can
+be obtained without caprice, however, by taking into consideration the
+following demands:
+
+(_a_) The required generalisation must likewise satisfy the general
+postulate of relativity.
+
+
+(_b_) If there is any matter in the domain under consideration, only
+its inertial mass, and thus according to Section XV only its energy is
+of importance for its effect in exciting a field.
+
+
+(_c_) Gravitational field and matter together must satisfy the law of
+the conservation of energy (and of impulse).
+
+
+Finally, the general principle of relativity permits us to determine
+the influence of the gravitational field on the course of all those
+processes which take place according to known laws when a gravitational
+field is absent _i.e._ which have already been fitted into the frame of
+the special theory of relativity. In this connection we proceed in
+principle according to the method which has already been explained for
+measuring-rods, clocks and freely moving material points.
+
+The theory of gravitation derived in this way from the general
+postulate of relativity excels not only in its beauty; nor in removing
+the defect attaching to classical mechanics which was brought to light
+in Section XXI; nor in interpreting the empirical law of the equality
+of inertial and gravitational mass; but it has also already explained a
+result of observation in astronomy, against which classical mechanics
+is powerless.
+
+If we confine the application of the theory to the case where the
+gravitational fields can be regarded as being weak, and in which all
+masses move with respect to the coordinate system with velocities which
+are small compared with the velocity of light, we then obtain as a
+first approximation the Newtonian theory. Thus the latter theory is
+obtained here without any particular assumption, whereas Newton had to
+introduce the hypothesis that the force of attraction between mutually
+attracting material points is inversely proportional to the square of
+the distance between them. If we increase the accuracy of the
+calculation, deviations from the theory of Newton make their
+appearance, practically all of which must nevertheless escape the test
+of observation owing to their smallness.
+
+We must draw attention here to one of these deviations. According to
+Newton’s theory, a planet moves round the sun in an ellipse, which
+would permanently maintain its position with respect to the fixed
+stars, if we could disregard the motion of the fixed stars themselves
+and the action of the other planets under consideration. Thus, if we
+correct the observed motion of the planets for these two influences,
+and if Newton’s theory be strictly correct, we ought to obtain for the
+orbit of the planet an ellipse, which is fixed with reference to the
+fixed stars. This deduction, which can be tested with great accuracy,
+has been confirmed for all the planets save one, with the precision
+that is capable of being obtained by the delicacy of observation
+attainable at the present time. The sole exception is Mercury, the
+planet which lies nearest the sun. Since the time of Leverrier, it has
+been known that the ellipse corresponding to the orbit of Mercury,
+after it has been corrected for the influences mentioned above, is not
+stationary with respect to the fixed stars, but that it rotates
+exceedingly slowly in the plane of the orbit and in the sense of the
+orbital motion. The value obtained for this rotary movement of the
+orbital ellipse was 43 seconds of arc per century, an amount ensured to
+be correct to within a few seconds of arc. This effect can be explained
+by means of classical mechanics only on the assumption of hypotheses
+which have little probability, and which were devised solely for this
+purponse.
+
+On the basis of the general theory of relativity, it is found that the
+ellipse of every planet round the sun must necessarily rotate in the
+manner indicated above; that for all the planets, with the exception of
+Mercury, this rotation is too small to be detected with the delicacy of
+observation possible at the present time; but that in the case of
+Mercury it must amount to 43 seconds of arc per century, a result which
+is strictly in agreement with observation.
+
+Apart from this one, it has hitherto been possible to make only two
+deductions from the theory which admit of being tested by observation,
+to wit, the curvature of light rays by the gravitational field of the
+sun,[22] and a displacement of the spectral lines of light reaching us
+from large stars, as compared with the corresponding lines for light
+produced in an analogous manner terrestrially (_i.e._ by the same kind
+of atom).[23] These two deductions from the theory have both been
+confirmed.
+
+
+ [22] First observed by Eddington and others in 1919. (Cf. Appendix
+ III).
+
+
+ [23] Established by Adams in 1924. (Cf. p. 132)
+
+
+PART III: CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
+
+
+XXX.
+
+COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY
+
+Part from the difficulty discussed in Section XXI, there is a second
+fundamental difficulty attending classical celestial mechanics, which,
+to the best of my knowledge, was first discussed in detail by the
+astronomer Seeliger. If we ponder over the question as to how the
+universe, considered as a whole, is to be regarded, the first answer
+that suggests itself to us is surely this: As regards space (and time)
+the universe is infinite. There are stars everywhere, so that the
+density of matter, although very variable in detail, is nevertheless on
+the average everywhere the same. In other words: However far we might
+travel through space, we should find everywhere an attenuated swarm of
+fixed stars of approrimately the same kind and density.
+
+This view is not in harmony with the theory of Newton. The latter
+theory rather requires that the universe should have a kind of centre
+in which the density of the stars is a maximum, and that as we proceed
+outwards from this centre the group-density of the stars should
+diminish, until finally, at great distances, it is succeeded by an
+infinite region of emptiness. The stellar universe ought to be a finite
+island in the infinite ocean of space.[24]
+
+
+ [24] _Proof_—According to the theory of Newton, the number of “lines
+ of force” which come from infinity and terminate in a mass m is
+ proportional to the mass _m_. If, on the average, the mass density ρ0
+ is constant throughout the universe, then a sphere of volume _V_ will
+ enclose the average mass ρ0_V_. Thus the number of lines of force
+ passing through the surface _F_ of the sphere into its interior is
+ proportional to ρ0_V_. For unit area of the surface of the sphere the
+ number of lines of force which enters the sphere is thus proportional
+ to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the
+ surface would ultimately become infinite with increasing radius _R_ of
+ the sphere, which is impossible.
+
+
+This conception is in itself not very satisfactory. It is still less
+satisfactory because it leads to the result that the light emitted by
+the stars and also individual stars of the stellar system are
+perpetually passing out into infinite space, never to return, and
+without ever again coming into interaction with other objects of
+nature. Such a finite material universe would be destined to become
+gradually but systematically impoverished.
+
+In order to escape this dilemma, Seeliger suggested a modification of
+Newton’s law, in which he assumes that for great distances the force of
+attraction between two masses diminishes more rapidly than would result
+from the inverse square law. In this way it is possible for the mean
+density of matter to be constant everywhere, even to infinity, without
+infinitely large gravitational fields being produced. We thus free
+ourselves from the distasteful conception that the material universe
+ought to possess something of the nature of a centre. Of course we
+purchase our emancipation from the fundamental difficulties mentioned,
+at the cost of a modification and complication of Newton’s law which
+has neither empirical nor theoretical foundation. We can imagine
+innumerable laws which would serve the same purpose, without our being
+able to state a reason why one of them is to be preferred to the
+others; for any one of these laws would be founded just as little on
+more general theoretical principles as is the law of Newton.
+
+
+XXXI.
+
+THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE
+
+But speculations on the structure of the universe also move in quite
+another direction. The development of non-Euclidean geometry led to the
+recognition of the fact, that we can cast doubt on the _infiniteness_
+of our space without coming into conflict with the laws of thought or
+with experience (Riemann, Helmholtz). These questions have already been
+treated in detail and with unsurpassable lucidity by Helmholtz and
+Poincaré, whereas I can only touch on them briefly here.
+
+In the first place, we imagine an existence in two dimensional space.
+Flat beings with flat implements, and in particular flat rigid
+measuring-rods, are free to move in a _plane_. For them nothing exists
+outside of this plane: that which they observe to happen to themselves
+and to their flat “things” is the all-inclusive reality of their plane.
+In particular, the constructions of plane Euclidean geometry can be
+carried out by means of the rods _e.g._ the lattice construction,
+considered in Section XXIV. In contrast to ours, the universe of these
+beings is two-dimensional; but, like ours, it extends to infinity. In
+their universe there is room for an infinite number of identical
+squares made up of rods, _i.e._ its volume (surface) is infinite. If
+these beings say their universe is “plane,” there is sense in the
+statement, because they mean that they can perform the constructions of
+plane Euclidean geometry with their rods. In this connection the
+individual rods always represent the same distance, independently of
+their position.
+
+Let us consider now a second two-dimensional existence, but this time
+on a spherical surface instead of on a plane. The flat beings with
+their measuring-rods and other objects fit exactly on this surface and
+they are unable to leave it. Their whole universe of observation
+extends exclusively over the surface of the sphere. Are these beings
+able to regard the geometry of their universe as being plane geometry
+and their rods withal as the realisation of “distance”? They cannot do
+this. For if they attempt to realise a straight line, they will obtain
+a curve, which we “three-dimensional beings” designate as a great
+circle, _i.e._ a self-contained line of definite finite length, which
+can be measured up by means of a measuring-rod. Similarly, this
+universe has a finite area that can be compared with the area, of a
+square constructed with rods. The great charm resulting from this
+consideration lies in the recognition of the fact that _the universe of
+these beings is finite and yet has no limits._
+
+But the spherical-surface beings do not need to go on a world-tour in
+order to perceive that they are not living in a Euclidean universe.
+They can convince themselves of this on every part of their “world,”
+provided they do not use too small a piece of it. Starting from a
+point, they draw “straight lines” (arcs of circles as judged in three
+dimensional space) of equal length in all directions. They will call
+the line joining the free ends of these lines a “circle.” For a plane
+surface, the ratio of the circumference of a circle to its diameter,
+both lengths being measured with the same rod, is, according to
+Euclidean geometry of the plane, equal to a constant value π, which is
+independent of the diameter of the circle. On their spherical surface
+our flat beings would find for this ratio the value
+
+image036
+
+
+_i.e._ a smaller value than π, the difference being the more
+considerable, the greater is the radius of the circle in comparison
+with the radius _R_ of the “world-sphere.” By means of this relation
+the spherical beings can determine the radius of their universe
+(“world”), even when only a relatively small part of their worldsphere
+is available for their measurements. But if this part is very small
+indeed, they will no longer be able to demonstrate that they are on a
+spherical “world” and not on a Euclidean plane, for a small part of a
+spherical surface differs only slightly from a piece of a plane of the
+same size.
+
+Thus if the spherical surface beings are living on a planet of which
+the solar system occupies only a negligibly small part of the spherical
+universe, they have no means of determining whether they are living in
+a finite or in an infinite universe, because the “piece of universe” to
+which they have access is in both cases practically plane, or
+Euclidean. It follows directly from this discussion, that for our
+sphere-beings the circumference of a circle first increases with the
+radius until the “circumference of the universe” is reached, and that
+it thenceforward gradually decreases to zero for still further
+increasing values of the radius. During this process the area of the
+circle continues to increase more and more, until finally it becomes
+equal to the total area of the whole “world-sphere.”
+
+Perhaps the reader will wonder why we have placed our “beings” on a
+sphere rather than on another closed surface. But this choice has its
+justification in the fact that, of all closed surfaces, the sphere is
+unique in possessing the property that all points on it are equivalent.
+I admit that the ratio of the circumference _c_ of a circle to its
+radius _r_ depends on _r_, but for a given value of _r_ it is the same
+for all points of the “worldsphere”; in other words, the “world-sphere”
+is a “surface of constant curvature.”
+
+To this two-dimensional sphere-universe there is a three-dimensional
+analogy, namely, the three-dimensional spherical space which was
+discovered by Riemann. its points are likewise all equivalent. It
+possesses a finite volume, which is determined by its “radius”
+(2π2_R_3). Is it possible to imagine a spherical space? To imagine a
+space means nothing else than that we imagine an epitome of our “space”
+experience, _i.e._ of experience that we can have in the movement of
+“rigid” bodies. In this sense we _can_ imagine a spherical space.
+
+Suppose we draw lines or stretch strings in all directions from a
+point, and mark off from each of these the distance _r_ with a
+measuring-rod. All the free end-points of these lengths lie on a
+spherical surface. We can specially measure up the area (_F_) of this
+surface by means of a square made up of measuring-rods. If the universe
+is Euclidean, then _F_ = 4π_r_2; if it is spherical, then _F_ is always
+less than 4π_r_2. With increasing values of _r, F_ increases from zero
+up to a maximum value which is determined by the “world-radius,” but
+for still further increasing values of _r_, the area gradually
+diminishes to zero. At first, the straight lines which radiate from the
+starting point diverge farther and farther from one another, but later
+they approach each other, and finally they run together again at a
+“counter-point” to the starting point. Under such conditions they have
+traversed the whole spherical space. It is easily seen that the
+three-dimensional spherical space is quite analogous to the
+two-dimensional spherical surface. It is finite (_i.e._ of finite
+volume), and has no bounds.
+
+It may be mentioned that there is yet another kind of curved space:
+“elliptical space.” It can be regarded as a curved space in which the
+two “counter-points” are identical (indistinguishable from each other).
+An elliptical universe can thus be considered to some extent as a
+curved universe possessing central symmetry.
+
+It follows from what has been said, that closed spaces without limits
+are conceivable. From amongst these, the spherical space (and the
+elliptical) excels in its simplicity, since all points on it are
+equivalent. As a result of this discussion, a most interesting question
+arises for astronomers and physicists, and that is whether the universe
+in which we live is infinite, or whether it is finite in the manner of
+the spherical universe. Our experience is far from being sufficient to
+enable us to answer this question. But the general theory of relativity
+permits of our answering it with a moderate degree of certainty, and in
+this connection the difficulty mentioned in Section XXX finds its
+solution.
+
+
+XXXII.
+
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
+
+According to the general theory of relativity, the geometrical
+properties of space are not independent, but they are determined by
+matter. Thus we can draw conclusions about the geometrical structure of
+the universe only if we base our considerations on the state of the
+matter as being something that is known. We know from experience that,
+for a suitably chosen co-ordinate system, the velocities of the stars
+are small as compared with the velocity of transmission of light. We
+can thus as a rough approximation arrive at a conclusion as to the
+nature of the universe as a whole, if we treat the matter as being at
+rest.
+
+We already know from our previous discussion that the behaviour of
+measuring-rods and clocks is influenced by gravitational fields, _i.e._
+by the distribution of matter. This in itself is sufficient to exclude
+the possibility of the exact validity of Euclidean geometry in our
+universe. But it is conceivable that our universe differs only slightly
+from a Euclidean one, and this notion seems all the more probable,
+since calculations show that the metrics of surrounding space is
+influenced only to an exceedingly small extent by masses even of the
+magnitude of our sun. We might imagine that, as regards geometry, our
+universe behaves analogously to a surface which is irregularly curved
+in its individual parts, but which nowhere departs appreciably from a
+plane: something like the rippled surface of a lake. Such a universe
+might fittingly be called a quasi-Euclidean universe. As regards its
+space it would be infinite. But calculation shows that in a
+quasi-Euclidean universe the average density of matter would
+necessarily be _nil_. Thus such a universe could not be inhabited by
+matter everywhere; it would present to us that unsatisfactory picture
+which we portrayed in Section XXX.
+
+If we are to have in the universe an average density of matter which
+differs from zero, however small may be that difference, then the
+universe cannot be quasi-Euclidean. On the contrary, the results of
+calculation indicate that if matter be distributed uniformly, the
+universe would necessarily be spherical (or elliptical). Since in
+reality the detailed distribution of matter is not uniform, the real
+universe will deviate in individual parts from the spherical, _i.e._
+the universe will be quasi-spherical. But it will be necessarily
+finite. In fact, the theory supplies us with a simple connection[25]
+between the space-expanse of the universe and the average density of
+matter in it.
+
+
+ [25] For the radius _R_ of the universe we obtain the equation
+
+
+image037
+
+
+The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027;
+ρ is the average density of the matter and _k_ is a constant connected
+with the Newtonian constant of gravitation.
+
+
+APPENDICES
+
+
+APPENDIX I
+
+SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
+
+(SUPPLEMENTARY TO SECTION XI)
+
+For the relative orientation of the co-ordinate systems indicated in
+Fig. 2, the _x_-axes of both systems permanently coincide. In the
+present case we can divide the problem into parts by considering first
+only events which are localised on the _x_-axis. Any such event is
+represented with respect to the co-ordinate system _K_ by the abscissa
+_x_ and the time _t_, and with respect to the system _K′_ by the
+abscissa _x′_ and the time _t′_. We require to find _x′_ and _t′_ when
+_x_ and _t_ are given.
+
+A light-signal, which is proceeding along the positive axis of _x_, is
+transmitted according to the equation
+
+_x_ = _ct_
+
+
+or
+
+_x_ – _ct_ = 0 . . . . . (1).
+
+
+Since the same light-signal has to be transmitted relative to _K′_ with
+the velocity _c_, the propagation relative to the system _K′_ will be
+represented by the analogous formula
+
+_x′_ – _ct′_ = 0 . . . . . (2)
+
+
+Those space-time points (events) which satisfy (1) must also satisfy
+(2). Obviously this will be the case when the relation
+
+(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).
+
+
+is fulfilled in general, where λ indicates a constant; for, according
+to (3), the disappearance of (_x_ – _ct_) involves the disappearance of
+(_x′_ – _ct′_).
+
+If we apply quite similar considerations to light rays which are being
+transmitted along the negative _x_-axis, we obtain the condition
+
+(_x′_ + _ct′_) = (_x + ct_) . . . (4).
+
+
+By adding (or subtracting) equations (3) and (4), and introducing for
+convenience the constants _a_ and _b_ in place of the constants λ and μ
+where
+
+image038
+
+
+and
+
+image039
+
+
+we obtain the equations
+
+image040
+
+
+We should thus have the solution of our problem, if the constants _a_
+and _b_ were known. These result from the following discussion.
+
+For the origin of _K′_ we have permanently _x′_ = 0, and hence
+according to the first of the equations (5)
+
+image041
+
+
+If we call _v_ the velocity with which the origin of _K′_ is moving
+relative to _K_, we then have
+
+image042
+
+
+The same value _v_ can be obtained from equations (5), if we calculate
+the velocity of another point of _K′_ relative to _K_, or the velocity
+(directed towards the negative _x_-axis) of a point of _K_ with respect
+to _K′_. In short, we can designate _v_ as the relative velocity of the
+two systems.
+
+Furthermore, the principle of relativity teaches us that, as judged
+from K, the length of a unit measuring-rod which is at rest with
+reference to _K′_ must be exactly the same as the length, as judged
+from _K′_, of a unit measuring-rod which is at rest relative to _K_. In
+order to see how the points of the _x′_-axis appear as viewed from _K_,
+we only require to take a “snapshot” of _K′_ from _K_; this means that
+we have to insert a particular value of _t_ (time of _K_), _e.g._ _t_ =
+0. For this value of _t_ we then obtain from the first of the equations
+(5)
+
+_x′_ = _ax_
+
+
+Two points of the _x′_-axis which are separated by the distance Δ_x′_ =
+1 when measured in the _K′_ system are thus separated in our
+instantaneous photograph by the distance
+
+image043
+
+
+But if the snapshot be taken from _K′_(_t′_ = 0), and if we eliminate
+_t_ from the equations (5), taking into account the expression (6), we
+obtain
+
+image044
+
+
+From this we conclude that two points on the _x_-axis separated by the
+distance 1 (relative to _K_) will be represented on our snapshot by the
+distance
+
+image045
+
+
+But from what has been said, the two snapshots must be identical; hence
+Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain
+
+image046
+
+
+The equations (6) and (7_b_) determine the constants _a_ and _b_. By
+inserting the values of these constants in (5), we obtain the first and
+the fourth of the equations given in Section XI.
+
+image047
+
+
+Thus we have obtained the Lorentz transformation for events on the
+_x_-axis. It satisfies the condition
+
+_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a).
+
+
+The extension of this result, to include events which take place
+outside the _x_-axis, is obtained by retaining equations (8) and
+supplementing them by the relations
+
+image048
+
+
+In this way we satisfy the postulate of the constancy of the velocity
+of light _in vacuo_ for rays of light of arbitrary direction, both for
+the system _K_ and for the system _K′_. This may be shown in the
+following manner.
+
+We suppose a light-signal sent out from the origin of _K_ at the time
+_t_ = 0. It will be propagated according to the equation
+
+image049
+
+
+or, if we square this equation, according to the equation
+
+_x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10).
+
+
+It is required by the law of propagation of light, in conjunction with
+the postulate of relativity, that the transmission of the signal in
+question should take place—as judged from _K′_—in accordance with the
+corresponding formula
+
+_r′_ = _ct′_
+
+
+or,
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).
+
+
+In order that equation (10_a_) may be a consequence of equation (10),
+we must have
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2)
+(11).
+
+
+Since equation (8_a_) must hold for points on the _x_-axis, we thus
+have σ = 1. It is easily seen that the Lorentz transformation really
+satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_)
+and (9), and hence also of (8) and (9). We have thus derived the
+Lorentz transformation.
+
+The Lorentz transformation represented by (8) and (9) still requires to
+be generalised. Obviously it is immaterial whether the axes of _K′_ be
+chosen so that they are spatially parallel to those of _K_. It is also
+not essential that the velocity of translation of _K′_ with respect to
+_K_ should be in the direction of the _x_-axis. A simple consideration
+shows that we are able to construct the Lorentz transformation in this
+general sense from two kinds of transformations, viz. from Lorentz
+transformations in the special sense and from purely spatial
+transformations. which corresponds to the replacement of the
+rectangular co-ordinate system by a new system with its axes pointing
+in other directions.
+
+Mathematically, we can characterise the generalised Lorentz
+transformation thus:
+
+It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions
+of _x, y, x, t_, of such a kind that the relation
+
+_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2
+(11_a_).
+
+
+is satisficd identically. That is to say: If we substitute their
+expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the
+left-hand side, then the left-hand side of (11_a_) agrees with the
+right-hand side.
+
+
+APPENDIX II
+
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)
+
+(SUPPLEMENTARY TO SECTION XVII)
+
+We can characterise the Lorentz transformation still more simply if we
+introduce the imaginary
+
+image031
+
+
+in place of _t_, as time-variable. If, in accordance with this, we
+insert
+
+image050
+
+
+and similarly for the accented system _K′_, then the condition which is
+identically satisfied by the transformation can be expressed thus:
+
+_x_1′2 + _x_2′2 + _x_3′2 + _x_4′2 = _x_12 + _x_22 + _x_32 + _x_42 (12).
+
+
+That is, by the afore-mentioned choice of “coordinates,” (11_a_) [see
+the end of Appendix II] is transformed into this equation.
+
+We see from (12) that the imaginary time co-ordinate _x_4, enters into
+the condition of transformation in exactly the same way as the space
+co-ordinates _x_1, _x_2, _x_3. It is due to this fact that, according
+to the theory of relativity, the “time” _x_4, enters into natural laws
+in the same form as the space co ordinates _x_1, _x_2, _x_3.
+
+A four-dimensional continuum described by the “co-ordinates” _x_1,
+_x_2, _x_3, _x_4, was called “world” by Minkowski, who also termed a
+point-event a “world-point.” From a “happening” in three-dimensional
+space, physics becomes, as it were, an “existence” in the
+four-dimensional “world.”
+
+This four-dimensional “world” bears a close similarity to the
+three-dimensional “space” of (Euclidean) analytical geometry. If we
+introduce into the latter a new Cartesian co-ordinate system (_x′_1,
+_x′_2, _x′_3) with the same origin, then _x′_1, _x′_2, _x′_3, are
+linear homogeneous functions of _x_1, _x_2, _x_3 which identically
+satisfy the equation
+
+_x_1′2 + _x_2′2 + _x_3′2 = _x_12 + _x_22 + _x_32
+
+
+The analogy with (12) is a complete one. We can regard Minkowski’s
+“world” in a formal manner as a four-dimensional Euclidean space (with
+an imaginary time coordinate); the Lorentz transformation corresponds
+to a “rotation” of the co-ordinate system in the four-dimensional
+“world.”
+
+
+APPENDIX III
+
+THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
+
+From a systematic theoretical point of view, we may imagine the process
+of evolution of an empirical science to be a continuous process of
+induction. Theories are evolved and are expressed in short compass as
+statements of a large number of individual observations in the form of
+empirical laws, from which the general laws can be ascertained by
+comparison. Regarded in this way, the development of a science bears
+some resemblance to the compilation of a classified catalogue. It is,
+as it were, a purely empirical enterprise.
+
+But this point of view by no means embraces the whole of the actual
+process; for it slurs over the important part played by intuition and
+deductive thought in the development of an exact science. As soon as a
+science has emerged from its initial stages, theoretical advances are
+no longer achieved merely by a process of arrangement. Guided by
+empirical data, the investigator rather develops a system of thought
+which, in general, is built up logically from a small number of
+fundamental assumptions, the so-called axioms. We call such a system of
+thought a _theory_. The theory finds the justification for its
+existence in the fact that it correlates a large number of single
+observations, and it is just here that the “truth” of the theory lies.
+
+Corresponding to the same complex of empirical data, there may be
+several theories, which differ from one another to a considerable
+extent. But as regards the deductions from the theories which are
+capable of being tested, the agreement between the theories may be so
+complete that it becomes difficult to find any deductions in which the
+two theories differ from each other. As an example, a case of general
+interest is available in the province of biology, in the Darwinian
+theory of the development of species by selection in the struggle for
+existence, and in the theory of development which is based on the
+hypothesis of the hereditary transmission of acquired characters.
+
+We have another instance of far-reaching agreement between the
+deductions from two theories in Newtonian mechanics on the one hand,
+and the general theory of relativity on the other. This agreement goes
+so far, that up to the present we have been able to find only a few
+deductions from the general theory of relativity which are capable of
+investigation, and to which the physics of pre-relativity days does not
+also lead, and this despite the profound difference in the fundamental
+assumptions of the two theories. In what follows, we shall again
+consider these important deductions, and we shall also discuss the
+empirical evidence appertaining to them which has hitherto been
+obtained.
+
+(_a_) Motion of the Perihelion of Mercury
+
+According to Newtonian mechanics and Newton’s law of gravitation, a
+planet which is revolving round the sun would describe an ellipse round
+the latter, or, more correctly, round the common centre of gravity of
+the sun and the planet. In such a system, the sun, or the common centre
+of gravity, lies in one of the foci of the orbital ellipse in such a
+manner that, in the course of a planet-year, the distance sun-planet
+grows from a minimum to a maximum, and then decreases again to a
+minimum. If instead of Newton’s law we insert a somewhat different law
+of attraction into the calculation, we find that, according to this new
+law, the motion would still take place in such a manner that the
+distance sun-planet exhibits periodic variations; but in this case the
+angle described by the line joining sun and planet during such a period
+(from perihelion—closest proximity to the sun—to perihelion) would
+differ from 360°. The line of the orbit would not then be a closed one
+but in the course of time it would fill up an annular part of the
+orbital plane, viz. between the circle of least and the circle of
+greatest distance of the planet from the sun.
+
+According also to the general theory of relativity, which differs of
+course from the theory of Newton, a small variation from the
+Newton-Kepler motion of a planet in its orbit should take place, and in
+such away, that the angle described by the radius sun-planet between
+one perhelion and the next should exceed that corresponding to one
+complete revolution by an amount given by
+
+image051
+
+
+(_N.B._—One complete revolution corresponds to the angle 2π in the
+absolute angular measure customary in physics, and the above expression
+given the amount by which the radius sun-planet exceeds this angle
+during the interval between one perihelion and the next.) In this
+expression _a_ represents the major semi-axis of the ellipse, _e_ its
+eccentricity, _c_ the velocity of light, and _T_ the period of
+revolution of the planet. Our result may also be stated as follows:
+According to the general theory of relativity, the major axis of the
+ellipse rotates round the sun in the same sense as the orbital motion
+of the planet. Theory requires that this rotation should amount to 43
+seconds of arc per century for the planet Mercury, but for the other
+Planets of our solar system its magnitude should be so small that it
+would necessarily escape detection.[26]
+
+
+ [26] Especially since the next planet Venus has an orbit that is
+ almost an exact circle, which makes it more difficult to locate the
+ perihelion with precision.
+
+
+In point of fact, astronomers have found that the theory of Newton does
+not suffice to calculate the observed motion of Mercury with an
+exactness corresponding to that of the delicacy of observation
+attainable at the present time. After taking account of all the
+disturbing influences exerted on Mercury by the remaining planets, it
+was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
+perihelial movement of the orbit of Mercury remained over, the amount
+of which does not differ sensibly from the above mentioned +43 seconds
+of arc per century. The uncertainty of the empirical result amounts to
+a few seconds only.
+
+(_b_) Deflection of Light by a Gravitational Field
+
+image052
+
+
+In Section XXII it has been already mentioned that according to the
+general theory of relativity, a ray of light will experience a
+curvature of its path when passing through a gravitational field, this
+curvature being similar to that experienced by the path of a body which
+is projected through a gravitational field. As a result of this theory,
+we should expect that a ray of light which is passing close to a
+heavenly body would be deviated towards the latter. For a ray of light
+which passes the sun at a distance of Δ sun-radii from its centre, the
+angle of deflection (α) should amount to
+
+image053
+
+
+It may be added that, according to the theory, half of this deflection
+is produced by the Newtonian field of attraction of the sun, and the
+other half by the geometrical modification (“curvature”) of space
+caused by the sun.
+
+This result admits of an experimental test by means of the photographic
+registration of stars during a total eclipse of the sun. The only
+reason why we must wait for a total eclipse is because at every other
+time the atmosphere is so strongly illuminated by the light from the
+sun that the stars situated near the sun’s disc are invisible. The
+predicted effect can be seen clearly from the accompanying diagram. If
+the sun (_S_) were not present, a star which is practically infinitely
+distant would be seen in the direction _D_1, as observed front the
+earth. But as a consequence of the deflection of light from the star by
+the sun, the star will be seen in the direction _D_2, _i.e._ at a
+somewhat greater distance from the centre of the sun than corresponds
+to its real position.
+
+In practice, the question is tested in the following way. The stars in
+the neighbourhood of the sun are photographed during a solar eclipse.
+
+In addition, a second photograph of the same stars is taken when the
+sun is situated at another position in the sky, _i.e._ a few months
+earlier or later. As compared with the standard photograph, the
+positions of the stars on the eclipse-photograph ought to appear
+displaced radially outwards (away from the centre of the sun) by an
+amount corresponding to the angle _a_.
+
+We are indebted to the [British] Royal Society and to the Royal
+Astronomical Society for the investigation of this important deduction.
+Undaunted by the [first world] war and by difficulties of both a
+material and a psychological nature aroused by the war, these societies
+equipped two expeditions—to Sobral (Brazil), and to the island of
+Principe (West Africa)—and sent several of Britain’s most celebrated
+astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to
+obtain photographs of the solar eclipse of 29th May, 1919. The relative
+discrepancies to be expected between the stellar photographs obtained
+during the eclipse and the comparison photographs amounted to a few
+hundredths of a millimetre only. Thus great accuracy was necessary in
+making the adjustments required for the taking of the photographs, and
+in their subsequent measurement.
+
+The results of the measurements confirmed the theory in a thoroughly
+satisfactory manner. The rectangular components of the observed and of
+the calculated deviations of the stars (in seconds of arc) are set
+forth in the following table of results:
+
+image054
+
+
+(_c_) Displacement of Spectral Lines Towards the Red
+
+In Section XXIII it has been shown that in a system _K′_ which is in
+rotation with regard to a Galileian system _K_, clocks of identical
+construction, and which are considered at rest with respect to the
+rotating reference-body, go at rates which are dependent on the
+positions of the clocks. We shall now examine this dependence
+quantitatively. A clock, which is situated at a distance r from the
+centre of the disc, has a velocity relative to _K_ which is given by
+
+_v_ = ω_r_,
+
+
+where ω represents the angular velocity of rotation of the disc _K′_
+with respect to _K_. If _v_0, represents the number of ticks of the
+clock per unit time (“rate” of the clock) relative to _K_ when the
+clock is at rest, then the “rate” of the clock (_v_) when it is moving
+relative to _K_ with a velocity _v_, but at rest with respect to the
+disc, will, in accordance with Section XII, be given by
+
+image055
+
+
+or with sufficient accuracy by
+
+image056
+
+
+This expression may also be stated in the following form:
+
+image057
+
+
+If we represent the difference of potential of the centrifugal force
+between the position of the clock and the centre of the disc by φ,
+_i.e._ the work, considered negatively, which must be performed on the
+unit of mass against the centrifugal force in order to transport it
+from the position of the clock on the rotating disc to the centre of
+the disc, then we have
+
+image058
+
+
+From this it follows that
+
+image059
+
+
+In the first place, we see from this expression that two clocks of
+identical construction will go at different rates when situated at
+different distances from the centre of the disc. This result is also
+valid from the standpoint of an observer who is rotating with the disc.
+
+Now, as judged from the disc, the latter is in a gravitational field of
+potential φ, hence the result we have obtained will hold quite
+generally for gravitational fields. Furthermore, we can regard an atom
+which is emitting spectral lines as a clock, so that the following
+statement will hold:
+
+_An atom absorbs or emits light of a frequency which is dependent on
+the potential of the gravitational field in which it is situated._
+
+The frequency of an atom situated on the surface of a heavenly body
+will be somewhat less than the frequency of an atom of the same element
+which is situated in free space (or on the surface of a smaller
+celestial body).
+
+Now φ = – _K (M/r)_, where _K_ is Newton’s constant of gravitation, and
+_M_ is the mass of the heavenly body. Thus a displacement towards the
+red ought to take place for spectral lines produced at the surface of
+stars as compared with the spectral lines of the same element produced
+at the surface of the earth, the amount of this displacement being
+
+image060
+
+
+For the sun, the displacement towards the red predicted by theory
+amounts to about two millionths of the wave-length. A trustworthy
+calculation is not possible in the case of the stars, because in
+general neither the mass _M_ nor the radius _r_ are known.
+
+It is an open question whether or not this effect exists, and at the
+present time (1920) astronomers are working with great zeal towards the
+solution. Owing to the smallness of the effect in the case of the sun,
+it is difficult to form an opinion as to its existence. Whereas Grebe
+and Bachem (Bonn), as a result of their own measurements and those of
+Evershed and Schwarzschild on the cyanogen bands, have placed the
+existence of the effect almost beyond doubt, while other investigators,
+particularly St. John, have been led to the opposite opinion in
+consequence of their measurements.
+
+Mean displacements of lines towards the less refrangible end of the
+spectrum are certainly revealed by statistical investigations of the
+fixed stars; but up to the present the examination of the available
+data does not allow of any definite decision being arrived at, as to
+whether or not these displacements are to be referred in reality to the
+effect of gravitation. The results of observation have been collected
+together, and discussed in detail from the standpoint of the question
+which has been engaging our attention here, in a paper by E. Freundlich
+entitled “Zur Prüfung der allgemeinen Relativitäts-Theorie” (_Die
+Naturwissenschaften_, 1919, No. 35, p. 520: Julius Springer, Berlin).
+
+At all events, a definite decision will be reached during the next few
+years. If the displacement of spectral lines towards the red by the
+gravitational potential does not exist, then the general theory of
+relativity will be untenable. On the other hand, if the cause of the
+displacement of spectral lines be definitely traced to the
+gravitational potential, then the study of this displacement will
+furnish us with important information as to the mass of the heavenly
+bodies.[27]
+
+
+ [27] The displacement of spectral lines towards the red end of the
+ spectrum was definitely established by Adams in 1924, by observations
+ on the dense companion of Sirius, for which the effect is about thirty
+ times greater than for the Sun. R.W.L.—translator
+
+
+APPENDIX IV
+
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
+
+(SUPPLEMENTARY TO SECTION XXXII)
+
+Since the publication of the first edition of this little book, our
+knowledge about the structure of space in the large (“cosmological
+problem”) has had an important development, which ought to be mentioned
+even in a popular presentation of the subject.
+
+My original considerations on the subject were based on two hypotheses:
+
+(1) There exists an average density of matter in the whole of space
+which is everywhere the same and different from zero.
+
+(2) The magnitude (“radius”) of space is independent of time.
+
+Both these hypotheses proved to be consistent, according to the general
+theory of relativity, but only after a hypothetical term was added to
+the field equations, a term which was not required by the theory as
+such nor did it seem natural from a theoretical point of view
+(“cosmological term of the field equations”).
+
+Hypothesis (2) appeared unavoidable to me at the time, since I thought
+that one would get into bottomless speculations if one departed from
+it.
+
+However, already in the ’twenties, the Russian mathematician Friedman
+showed that a different hypothesis was natural from a purely
+theoretical point of view. He realized that it was possible to preserve
+hypothesis (1) without introducing the less natural cosmological term
+into the field equations of gravitation, if one was ready to drop
+hypothesis (2). Namely, the original field equations admit a solution
+in which the “world radius” depends on time (expanding space). In that
+sense one can say, according to Friedman, that the theory demands an
+expansion of space.
+
+A few years later Hubble showed, by a special investigation of the
+extra-galactic nebulae (“milky ways”), that the spectral lines emitted
+showed a red shift which increased regularly with the distance of the
+nebulae. This can be interpreted in regard to our present knowledge
+only in the sense of Doppler’s principle, as an expansive motion of the
+system of stars in the large—as required, according to Friedman, by the
+field equations of gravitation. Hubble’s discovery can, therefore, be
+considered to some extent as a confirmation of the theory.
+
+There does arise, however, a strange difficulty. The interpretation of
+the galactic line-shift discovered by Hubble as an expansion (which can
+hardly be doubted from a theoretical point of view), leads to an origin
+of this expansion which lies “only” about 109 years ago, while physical
+astronomy makes it appear likely that the development of individual
+stars and systems of stars takes considerably longer. It is in no way
+known how this incongruity is to be overcome.
+
+I further want to remark that the theory of expanding space, together
+with the empirical data of astronomy, permit no decision to be reached
+about the finite or infinite character of (three-dimensional) space,
+while the original “static” hypothesis of space yielded the closure
+(finiteness) of space.
+
+_K_ = co-ordinate system
+
+_x, y_ = two-dimensional co-ordinates
+
+_x, y, z_ = three-dimensional co-ordinates
+
+_x, y, z, t_ = four-dimensional co-ordinates
+
+_t_ = time
+
+_I_ = distance
+
+_v_ = velocity
+
+_F_ = force
+
+_G_ = gravitational field
+
+
+
+
+*** END OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***
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+<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of Relativity: The Special and General Theory, by Albert Einstein</div>
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+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: Relativity: The Special and General Theory</div>
+<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: Albert Einstein</div>
+<div style='display:block; margin:1em 0'>Release Date: October 1, 2009 [eBook #30155]<br />
+[Most recently updated: May 2, 2023]</div>
+<div style='display:block; margin:1em 0'>Language: English</div>
+<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Robert Hux</div>
+<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK RELATIVITY: THE SPECIAL AND GENERAL THEORY ***</div>
+
+<div class="fig" style="width:70%;">
+<img src="images/cover.jpg" style="width:100%;" alt="cover " /><br/><br/>
+</div>
+
+<h1>Relativity: The Special and General Theory</h1>
+
+<h2>by Albert Einstein</h2>
+
+<h4>Authorised Translation by Robert W. Lawson</h4>
+<hr />
+
+<p>ALBERT EINSTEIN REFERENCE ARCHIVE</p>
+
+<p>RELATIVITY: THE SPECIAL AND GENERAL THEORY</p>
+
+<p>BY ALBERT EINSTEIN<br/><br/></p>
+
+<p>Written: 1916 (this revised edition: 1924)</p>
+
+<p>Source: Relativity: The Special and General Theory (1920)</p>
+
+<p>Publisher: Methuen &amp; Co Ltd</p>
+
+<p>First Published: December, 1916</p>
+
+<p>Translated: Robert W. Lawson (Authorised translation)</p>
+
+<p>Transcription/Markup: Brian Basgen</p>
+
+<p>Transcription to text: Gregory B. Newby</p>
+
+<p>Thanks to: Einstein Reference Archive (marxists.org)</p>
+
+<p>The Einstein Reference Archive is online at:</p>
+
+<p>http://www.marxists.org/reference/archive/einstein/index.htm<br/><br/></p>
+
+<h3>Contents</h3>
+
+<table summary="">
+
+<tr>
+<td> <a href="#pref01">Preface</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part01"><b>Part I: The Special Theory of Relativity</b></a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap01">I. Physical Meaning of Geometrical Propositions</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap02">II. The System of Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap03">III. Space and Time in Classical Mechanics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap04">IV. The Galileian System of Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap05">V. The Principle of Relativity (in the Restricted Sense)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap06">VI. The Theorem of the Addition of Velocities employed in Classical Mechanics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap07">VII. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap08">VIII. On the Idea of Time in Physics</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap09">IX. The Relativity of Simultaneity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap10">X. On the Relativity of the Conception of Distance</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap11">XI. The Lorentz Transformation</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap12">XII. The Behaviour of Measuring-Rods and Clocks in Motion</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap13">XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap14">XIV. The Heuristic Value of the Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap15">XV. General Results of the Theory</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap16">XVI. Experience and the Special Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap17">XVII. Minkowski’s Four-dimensional Space</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part02"><b>Part II: The General Theory of Relativity</b></a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap18">XVIII. Special and General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap19">XIX. The Gravitational Field</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap20">XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap21">XXI. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap22">XXII. A Few Inferences from the General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap23">XXIII. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap24">XXIV. Euclidean and non-Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap25">XXV. Gaussian Co-ordinates</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap26">XXVI. The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap27">XXVII. The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap28">XXVIII. Exact Formulation of the General Principle of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap29">XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#part03"><b>Part III: Considerations on the Universe as a Whole</b></a></td>
+</tr>
+<tr>
+<td> <a href="#chap30">XXX. Cosmological Difficulties of Newton’s Theory</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap31">XXXI. The Possibility of a “Finite” and yet “Unbounded” Universe</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap32">XXXII. The Structure of Space According to the General Theory of Relativity</a><br/><br/></td>
+</tr>
+
+<tr>
+<td> <a href="#chap33">Appendices:</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap34">I. Simple Derivation of the Lorentz Transformation (supplementary to section XI)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap35">II. Minkowski’s Four-Dimensional Space (“World”) (supplementary to section XVII)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap36">III. The Experimental Confirmation of the General Theory of Relativity</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap37">IV. The Structure of Space According to the General Theory of Relativity (supplementary to section XXXII)</a></td>
+</tr>
+
+<tr>
+<td> <a href="#chap38">V. Relativity and the Problem of Space</a></td>
+</tr>
+
+</table>
+
+<p>
+Note: The fifth Appendix was added by Einstein at the time of the fifteenth
+re-printing of this book; and as a result is still under copyright restrictions
+so cannot be added without the permission of the publisher.
+</p>
+
+<div class="chapter">
+
+<h3><a name="pref01"></a>PREFACE</h3>
+
+<p>
+The present book is intended, as far as possible, to give an exact insight into
+the theory of Relativity to those readers who, from a general scientific and
+philosophical point of view, are interested in the theory, but who are not
+conversant with the mathematical apparatus of theoretical physics. The work
+presumes a standard of education corresponding to that of a university
+matriculation examination, and, despite the shortness of the book, a fair
+amount of patience and force of will on the part of the reader. The author has
+spared himself no pains in his endeavour to present the main ideas in the
+simplest and most intelligible form, and on the whole, in the sequence and
+connection in which they actually originated. In the interest of clearness, it
+appeared to me inevitable that I should repeat myself frequently, without
+paying the slightest attention to the elegance of the presentation. I adhered
+scrupulously to the precept of that brilliant theoretical physicist L.
+Boltzmann, according to whom matters of elegance ought to be left to the tailor
+and to the cobbler. I make no pretence of having withheld from the reader
+difficulties which are inherent to the subject. On the other hand, I have
+purposely treated the empirical physical foundations of the theory in a
+“step-motherly” fashion, so that readers unfamiliar with physics may
+not feel like the wanderer who was unable to see the forest for the trees. May
+the book bring some one a few happy hours of suggestive thought!
+</p>
+
+<p>
+December, 1916
+</p>
+
+<p class="right"> A. EINSTEIN
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part01"></a>PART I: THE SPECIAL THEORY OF RELATIVITY</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap01"></a>I.<br/>
+PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS</h3>
+
+<p>
+In your schooldays most of you who read this book made acquaintance with the
+noble building of Euclid’s geometry, and you remember—perhaps with more
+respect than love—the magnificent structure, on the lofty staircase of which
+you were chased about for uncounted hours by conscientious teachers. By reason
+of our past experience, you would certainly regard everyone with disdain who
+should pronounce even the most out-of-the-way proposition of this science to be
+untrue. But perhaps this feeling of proud certainty would leave you immediately
+if some one were to ask you: “What, then, do you mean by the assertion
+that these propositions are true?” Let us proceed to give this question a
+little consideration.
+</p>
+
+<p>
+Geometry sets out from certain conceptions such as “plane,”
+“point,” and “straight line,” with which we are able to
+associate more or less definite ideas, and from certain simple propositions
+(axioms) which, in virtue of these ideas, we are inclined to accept as
+“true.” Then, on the basis of a logical process, the justification of
+which we feel ourselves compelled to admit, all remaining propositions are
+shown to follow from those axioms, <i>i.e.</i> they are proven. A proposition is then
+correct (“true”) when it has been derived in the recognised manner
+from the axioms. The question of “truth” of the individual
+geometrical propositions is thus reduced to one of the “truth” of the
+axioms. Now it has long been known that the last question is not only
+unanswerable by the methods of geometry, but that it is in itself entirely
+without meaning. We cannot ask whether it is true that only one straight line
+goes through two points. We can only say that Euclidean geometry deals with
+things called “straight lines,” to each of which is ascribed the
+property of being uniquely determined by two points situated on it. The concept
+“true” does not tally with the assertions of pure geometry, because
+by the word “true” we are eventually in the habit of designating
+always the correspondence with a “real” object; geometry, however, is
+not concerned with the relation of the ideas involved in it to objects of
+experience, but only with the logical connection of these ideas among
+themselves.
+</p>
+
+<p>
+It is not difficult to understand why, in spite of this, we feel constrained to
+call the propositions of geometry “true.” Geometrical ideas
+correspond to more or less exact objects in nature, and these last are
+undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought
+to refrain from such a course, in order to give to its structure the largest
+possible logical unity. The practice, for example, of seeing in a
+“distance” two marked positions on a practically rigid body is
+something which is lodged deeply in our habit of thought. We are accustomed
+further to regard three points as being situated on a straight line, if their
+apparent positions can be made to coincide for observation with one eye, under
+suitable choice of our place of observation.
+</p>
+
+<p>
+If, in pursuance of our habit of thought, we now supplement the propositions of
+Euclidean geometry by the single proposition that two points on a practically
+rigid body always correspond to the same distance (line-interval),
+independently of any changes in position to which we may subject the body, the
+propositions of Euclidean geometry then resolve themselves into propositions on
+the possible relative position of practically rigid bodies.<a
+href="#linknote-1" name="linknoteref-1" id="linknoteref-1">[1]</a> Geometry
+which has been supplemented in this way is then to be treated as a branch of
+physics. We can now legitimately ask as to the “truth” of
+geometrical propositions interpreted in this way, since we are justified in
+asking whether these propositions are satisfied for those real things we have
+associated with the geometrical ideas. In less exact terms we can express this
+by saying that by the “truth” of a geometrical proposition in this
+sense we understand its validity for a construction with rule and compasses.
+</p>
+
+<p>
+<a name="linknote-1" id="linknote-1">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-1"> [1]</a><br/> It follows that a natural object is
+associated also with a straight line. Three points <i>A, B</i> and <i>C</i> on a rigid body
+thus lie in a straight line when the points <i>A</i> and <i>C</i> being given, <i>B</i> is chosen
+such that the sum of the distances <i>AB</i> and <i>BC</i> is as short as possible. This
+incomplete suggestion will suffice for the present purpose.
+</p>
+
+<p>
+Of course the conviction of the “truth” of geometrical propositions
+in this sense is founded exclusively on rather incomplete experience. For the
+present we shall assume the “truth” of the geometrical propositions,
+then at a later stage (in the general theory of relativity) we shall see that
+this “truth” is limited, and we shall consider the extent of its
+limitation.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap02"></a>II.<br/>
+THE SYSTEM OF CO-ORDINATES</h3>
+
+<p>
+On the basis of the physical interpretation of distance which has been
+indicated, we are also in a position to establish the distance between two
+points on a rigid body by means of measurements. For this purpose we require a
+“distance” (rod <i>S</i>) which is to be used once and for all, and
+which we employ as a standard measure. If, now, <i>A</i> and <i>B</i> are two
+points on a rigid body, we can construct the line joining them according to the
+rules of geometry; then, starting from <i>A</i>, we can mark off the distance
+<i>S</i> time after time until we reach <i>B</i>. The number of these
+operations required is the numerical measure of the distance <i>AB</i>. This is
+the basis of all measurement of length.<a href="#linknote-2"
+name="linknoteref-2" id="linknoteref-2">[2]</a>
+</p>
+
+<p>
+<a name="linknote-2" id="linknote-2">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-2"> [2]</a><br/> Here we have assumed that there is
+nothing left over <i>i.e.</i> that the measurement gives a whole number. This
+difficulty is got over by the use of divided measuring-rods, the introduction
+of which does not demand any fundamentally new method.
+</p>
+
+
+<p>
+Every description of the scene of an event or of the position of an object in
+space is based on the specification of the point on a rigid body (body of
+reference) with which that event or object coincides. This applies not only to
+scientific description, but also to everyday life. If I analyse the place
+specification “Trafalgar Square, London”<a href="#linknote-3" name="linknoteref-3" id="linknoteref-3">[3]</a> I arrive at
+the following result. The earth is the rigid body to which the specification of
+place refers; “Trafalgar Square, London” is a well-defined point, to
+which a name has been assigned, and with which the event coincides in
+space.<a href="#linknote-4" name="linknoteref-4" id="linknoteref-4">[4]</a>
+</p>
+
+<p>
+<a name="linknote-3" id="linknote-3">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-3"> [3]</a><br/> 
+I have chosen this as being more familiar to the English reader than the
+“Potzdammer Platz, Berlin,” which is referred to in the original.
+(R. W. L.)
+</p>
+
+<p>
+<a name="linknote-4" id="linknote-4">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-4"> [4]</a><br/> It is not necessary here to investigate
+further the significance of the expression “coincidence in space.”
+This conception is sufficiently obvious to ensure that differences of opinion
+are scarcely likely to arise as to its applicability in practice.
+</p>
+
+<p>
+This primitive method of place specification deals only with places on the
+surface of rigid bodies, and is dependent on the existence of points on this
+surface which are distinguishable from each other. But we can free ourselves
+from both of these limitations without altering the nature of our specification
+of position. If, for instance, a cloud is hovering over Trafalgar Square, then we
+can determine its position relative to the surface of the earth by erecting a
+pole perpendicularly on the Square, so that it reaches the cloud. The length of
+the pole measured with the standard measuring-rod, combined with the
+specification of the position of the foot of the pole, supplies us with a
+complete place specification. On the basis of this illustration, we are able to
+see the manner in which a refinement of the conception of position has been
+developed.
+</p>
+
+<p>
+(<i>a</i>) We imagine the rigid body, to which the place specification is referred,
+supplemented in such a manner that the object whose position we require is
+reached by the completed rigid body.
+</p>
+
+<p>
+(<i>b</i>) In locating the position of the object, we make use of a number (here the
+length of the pole measured with the measuring-rod) instead of designated
+points of reference.
+</p>
+
+<p>
+(<i>c</i>) We speak of the height of the cloud even when the pole which reaches the
+cloud has not been erected. By means of optical observations of the cloud from
+different positions on the ground, and taking into account the properties of
+the propagation of light, we determine the length of the pole we should have
+required in order to reach the cloud.
+</p>
+
+<p>
+From this consideration we see that it will be advantageous if, in the
+description of position, it should be possible by means of numerical measures
+to make ourselves independent of the existence of marked positions (possessing
+names) on the rigid body of reference. In the physics of measurement this is
+attained by the application of the Cartesian system of co-ordinates.
+</p>
+
+<p>
+This consists of three plane surfaces perpendicular to each other and rigidly
+attached to a rigid body. Referred to a system of co-ordinates, the scene of
+any event will be determined (for the main part) by the specification of the
+lengths of the three perpendiculars or co-ordinates (<i>x, y, z</i>) which can be
+dropped from the scene of the event to those three plane surfaces. The lengths
+of these three perpendiculars can be determined by a series of manipulations
+with rigid measuring-rods performed according to the rules and methods laid
+down by Euclidean geometry.
+</p>
+
+<p>
+In practice, the rigid surfaces which constitute the system of co-ordinates are
+generally not available; furthermore, the magnitudes of the co-ordinates are
+not actually determined by constructions with rigid rods, but by indirect
+means. If the results of physics and astronomy are to maintain their clearness,
+the physical meaning of specifications of position must always be sought in
+accordance with the above considerations.<a href="#linknote-5" name="linknoteref-5" id="linknoteref-5">[5]</a>
+</p>
+
+<p>
+<a name="linknote-5" id="linknote-5">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-5"> [5]</a><br/> A refinement and modification of these
+views does not become necessary until we come to deal with the general theory
+of relativity, treated in the second part of this book.
+</p>
+
+<p>
+We thus obtain the following result: Every description of events in space
+involves the use of a rigid body to which such events have to be referred. The
+resulting relationship takes for granted that the laws of Euclidean geometry
+hold for “distances;” the “distance” being represented
+physically by means of the convention of two marks on a rigid body.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap03"></a>III.<br/>
+SPACE AND TIME IN CLASSICAL MECHANICS</h3>
+
+<p>
+The purpose of mechanics is to describe how bodies change their position in
+space with “time.” I should load my conscience with grave sins
+against the sacred spirit of lucidity were I to formulate the aims of mechanics
+in this way, without serious reflection and detailed explanations. Let us
+proceed to disclose these sins.
+</p>
+
+<p>
+It is not clear what is to be understood here by “position” and
+“space.” I stand at the window of a railway carriage which is
+travelling uniformly, and drop a stone on the embankment, without throwing it.
+Then, disregarding the influence of the air resistance, I see the stone descend
+in a straight line. A pedestrian who observes the misdeed from the footpath
+notices that the stone falls to earth in a parabolic curve. I now ask: Do the
+“positions” traversed by the stone lie “in reality” on a
+straight line or on a parabola? Moreover, what is meant here by motion “in
+space”? From the considerations of the previous section the answer is
+self-evident. In the first place we entirely shun the vague word
+“space,” of which, we must honestly acknowledge, we cannot form the
+slightest conception, and we replace it by “motion relative to a
+practically rigid body of reference.” The positions relative to the body
+of reference (railway carriage or embankment) have already been defined in
+detail in the preceding section. If instead of “body of reference” we
+insert “system of co-ordinates,” which is a useful idea for
+mathematical description, we are in a position to say: The stone traverses a
+straight line relative to a system of co-ordinates rigidly attached to the
+carriage, but relative to a system of co-ordinates rigidly attached to the
+ground (embankment) it describes a parabola. With the aid of this example it is
+clearly seen that there is no such thing as an independently existing
+trajectory (lit. “path-curve”<a href="#linknote-6" name="linknoteref-6" id="linknoteref-6">[6]</a>, but only a trajectory
+relative to a particular body of reference.
+</p>
+
+<p>
+<a name="linknote-6" id="linknote-6">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-6"> [6]</a><br/> That is, a curve along which the body
+moves.
+</p>
+
+<p>
+In order to have a <i>complete</i> description of the motion, we must specify how the
+body alters its position <i>with time; i.e.</i> for every point on the trajectory it
+must be stated at what time the body is situated there. These data must be
+supplemented by such a definition of time that, in virtue of this definition,
+these time-values can be regarded essentially as magnitudes (results of
+measurements) capable of observation. If we take our stand on the ground of
+classical mechanics, we can satisfy this requirement for our illustration in
+the following manner. We imagine two clocks of identical construction; the man
+at the railway-carriage window is holding one of them, and the man on the
+footpath the other. Each of the observers determines the position on his own
+reference-body occupied by the stone at each tick of the clock he is holding in
+his hand. In this connection we have not taken account of the inaccuracy
+involved by the finiteness of the velocity of propagation of light. With this
+and with a second difficulty prevailing here we shall have to deal in detail
+later.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap04"></a>IV.<br/>THE GALILEIAN SYSTEM OF CO-ORDINATES</h3>
+
+
+<p>
+As is well known, the fundamental law of the mechanics of Galilei-Newton, which
+is known as the <i>law of inertia</i>, can be stated thus: A body removed sufficiently
+far from other bodies continues in a state of rest or of uniform motion in a
+straight line. This law not only says something about the motion of the bodies,
+but it also indicates the reference-bodies or systems of coordinates,
+permissible in mechanics, which can be used in mechanical description. The
+visible fixed stars are bodies for which the law of inertia certainly holds to
+a high degree of approximation. Now if we use a system of co-ordinates which is
+rigidly attached to the earth, then, relative to this system, every fixed star
+describes a circle of immense radius in the course of an astronomical day, a
+result which is opposed to the statement of the law of inertia. So that if we
+adhere to this law we must refer these motions only to systems of coordinates
+relative to which the fixed stars do not move in a circle. A system of
+co-ordinates of which the state of motion is such that the law of inertia holds
+relative to it is called a “Galileian system of co-ordinates.” The
+laws of the mechanics of Galilei-Newton can be regarded as valid only for a
+Galileian system of co-ordinates.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap05"></a>V.<br/>
+THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)</h3>
+
+<p>
+In order to attain the greatest possible clearness, let us return to our
+example of the railway carriage supposed to be travelling uniformly. We call
+its motion a uniform translation (“uniform” because it is of constant
+velocity and direction, “translation” because although the carriage
+changes its position relative to the embankment yet it does not rotate in so
+doing). Let us imagine a raven flying through the air in such a manner that its
+motion, as observed from the embankment, is uniform and in a straight line. If
+we were to observe the flying raven from the moving railway carriage. we should
+find that the motion of the raven would be one of different velocity and
+direction, but that it would still be uniform and in a straight line. Expressed
+in an abstract manner we may say: If a mass <i>m</i> is moving uniformly in a
+straight line with respect to a co-ordinate system <i>K</i>, then it will also be
+moving uniformly and in a straight line relative to a second co-ordinate system
+<i>K&#x2032;</i> provided that the latter is executing a uniform translatory motion with
+respect to <i>K</i>. In accordance with the discussion contained in the preceding
+section, it follows that:
+</p>
+
+<p>
+If <i>K</i> is a Galileian co-ordinate system. then every other co-ordinate system <i>K&#x2032;</i>
+is a Galileian one, when, in relation to <i>K</i>, it is in a condition of uniform
+motion of translation. Relative to <i>K&#x2032;</i> the mechanical laws of Galilei-Newton
+hold good exactly as they do with respect to <i>K</i>.
+</p>
+
+<p>
+We advance a step farther in our generalisation when we express the tenet thus:
+If, relative to <i>K</i>, <i>K&#x2032;</i> is a uniformly moving co-ordinate system devoid of
+rotation, then natural phenomena run their course with respect to <i>K&#x2032;</i> according
+to exactly the same general laws as with respect to <i>K</i>. This statement is called
+the <i>principle of relativity</i> (in the restricted sense).
+</p>
+
+<p>
+As long as one was convinced that all natural phenomena were capable of
+representation with the help of classical mechanics, there was no need to doubt
+the validity of this principle of relativity. But in view of the more recent
+development of electrodynamics and optics it became more and more evident that
+classical mechanics affords an insufficient foundation for the physical
+description of all natural phenomena. At this juncture the question of the
+validity of the principle of relativity became ripe for discussion, and it did
+not appear impossible that the answer to this question might be in the
+negative.
+</p>
+
+<p>
+Nevertheless, there are two general facts which at the outset speak very much
+in favour of the validity of the principle of relativity. Even though classical
+mechanics does not supply us with a sufficiently broad basis for the
+theoretical presentation of all physical phenomena, still we must grant it a
+considerable measure of “truth,” since it supplies us with the actual
+motions of the heavenly bodies with a delicacy of detail little short of
+wonderful. The principle of relativity must therefore apply with great accuracy
+in the domain of <i>mechanics</i>. But that a principle of such broad generality
+should hold with such exactness in one domain of phenomena, and yet should be
+invalid for another, is <i>a priori</i> not very probable.
+</p>
+
+<p>
+We now proceed to the second argument, to which, moreover, we shall return
+later. If the principle of relativity (in the restricted sense) does not hold,
+then the Galileian co-ordinate systems <i>K, K&#x2032;, K&#x2033;</i>, etc., which are moving
+uniformly relative to each other, will not be <i>equivalent</i> for the description of
+natural phenomena. In this case we should be constrained to believe that
+natural laws are capable of being formulated in a particularly simple manner,
+and of course only on condition that, from amongst all possible Galileian
+co-ordinate systems, we should have chosen <i>one</i> (<i>K<sub>0</sub></i>) of a particular
+state of motion as our body of reference. We should then be justified (because
+of its merits for the description of natural phenomena) in calling this system
+“absolutely at rest,” and all other Galileian systems <i>K</i>
+“in motion.” If, for instance, our embankment were the system
+<i>K<sub>0</sub></i> then our railway carriage would be a system <i>K</i>, relative to which
+less simple laws would hold than with respect to <i>K<sub>0</sub></i>. This diminished
+simplicity would be due to the fact that the carriage <i>K</i> would be in motion
+(<i>i.e.</i> “really”)with respect to <i>K<sub>0</sub></i>. In the general laws of
+nature which have been formulated with reference to <i>K</i>, the magnitude and
+direction of the velocity of the carriage would necessarily play a part. We
+should expect, for instance, that the note emitted by an organpipe placed with
+its axis parallel to the direction of travel would be different from that
+emitted if the axis of the pipe were placed perpendicular to this direction.
+</p>
+
+<p>
+Now in virtue of its motion in an orbit round the sun, our earth is comparable
+with a railway carriage travelling with a velocity of about 30 kilometres per
+second. If the principle of relativity were not valid we should therefore
+expect that the direction of motion of the earth at any moment would enter into
+the laws of nature, and also that physical systems in their behaviour would be
+dependent on the orientation in space with respect to the earth. For owing to
+the alteration in direction of the velocity of revolution of the earth in the
+course of a year, the earth cannot be at rest relative to the hypothetical
+system <i>K<sub>0</sub></i> throughout the whole year. However, the most careful
+observations have never revealed such anisotropic properties in terrestrial
+physical space, <i>i.e.</i> a physical non-equivalence of different directions. This
+is very powerful argument in favour of the principle of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap06"></a>VI.<br/>
+THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL MECHANICS</h3>
+
+<p>
+Let us suppose our old friend the railway carriage to be travelling along the
+rails with a constant velocity <i>v</i>, and that a man traverses the length of the
+carriage in the direction of travel with a velocity <i>w</i>. How quickly or, in other
+words, with what velocity <i>W</i> does the man advance relative to the embankment
+during the process? The only possible answer seems to result from the following
+consideration: If the man were to stand still for a second, he would advance
+relative to the embankment through a distance <i>v</i> equal numerically to the
+velocity of the carriage. As a consequence of his walking, however, he
+traverses an additional distance w relative to the carriage, and hence also
+relative to the embankment, in this second, the distance w being numerically
+equal to the velocity with which he is walking. Thus in total he covers the
+distance <i>W = v + w</i> relative to the embankment in the second considered. We shall
+see later that this result, which expresses the theorem of the addition of
+velocities employed in classical mechanics, cannot be maintained; in other
+words, the law that we have just written down does not hold in reality. For the
+time being, however, we shall assume its correctness.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap07"></a>VII.<br/>
+THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH THE
+PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+There is hardly a simpler law in physics than that according to which light is
+propagated in empty space. Every child at school knows, or believes he knows,
+that this propagation takes place in straight lines with a velocity <i>c</i> = 300,000
+km./sec. At all events we know with great exactness that this velocity is the
+same for all colours, because if this were not the case, the minimum of
+emission would not be observed simultaneously for different colours during the
+eclipse of a fixed star by its dark neighbour. By means of similar
+considerations based on observations of double stars, the Dutch astronomer De
+Sitter was also able to show that the velocity of propagation of light cannot
+depend on the velocity of motion of the body emitting the light. The assumption
+that this velocity of propagation is dependent on the direction “in
+space” is in itself improbable.
+</p>
+
+<p>
+In short, let us assume that the simple law of the constancy of the velocity of
+light <i>c</i> (in vacuum) is justifiably believed by the child at school. Who would
+imagine that this simple law has plunged the conscientiously thoughtful
+physicist into the greatest intellectual difficulties? Let us consider how
+these difficulties arise.
+</p>
+
+<p>
+Of course we must refer the process of the propagation of light (and indeed
+every other process) to a rigid reference-body (co-ordinate system). As such a
+system let us again choose our embankment. We shall imagine the air above it to
+have been removed. If a ray of light be sent along the embankment, we see from
+the above that the tip of the ray will be transmitted with the velocity <i>c</i>
+relative to the embankment. Now let us suppose that our railway carriage is
+again travelling along the railway lines with the velocity <i>v</i>, and that its
+direction is the same as that of the ray of light, but its velocity of course
+much less. Let us inquire about the velocity of propagation of the ray of light
+relative to the carriage. It is obvious that we can here apply the
+consideration of the previous section, since the ray of light plays the part of
+the man walking along relatively to the carriage. The velocity <i>W</i> of the man
+relative to the embankment is here replaced by the velocity of light relative
+to the embankment. <i>w</i> is the required velocity of light with respect to the
+carriage, and we have
+</p>
+
+<p>
+<i>w = c – v.</i>
+</p>
+
+<p>
+The velocity of propagation ot a ray of light relative to the carriage thus
+comes out smaller than <i>c</i>.
+</p>
+
+<p>
+But this result comes into conflict with the principle of relativity set forth
+in Section V. For, like every other general law of nature, the law of the
+transmission of light <i>in vacuo</i> [in vacuum] must, according to the principle of
+relativity, be the same for the railway carriage as reference-body as when the
+rails are the body of reference. But, from our above consideration, this would
+appear to be impossible. If every ray of light is propagated relative to the
+embankment with the velocity <i>c</i>, then for this reason it would appear that
+another law of propagation of light must necessarily hold with respect to the
+carriage—a result contradictory to the principle of relativity.
+</p>
+
+<p>
+In view of this dilemma there appears to be nothing else for it than to abandon
+either the principle of relativity or the simple law of the propagation of
+light <i>in vacuo</i>. Those of you who have carefully followed the preceding
+discussion are almost sure to expect that we should retain the principle of
+relativity, which appeals so convincingly to the intellect because it is so
+natural and simple. The law of the propagation of light <i>in vacuo</i> would then
+have to be replaced by a more complicated law conformable to the principle of
+relativity. The development of theoretical physics shows, however, that we
+cannot pursue this course. The epoch-making theoretical investigations of H. A.
+Lorentz on the electrodynamical and optical phenomena connected with moving
+bodies show that experience in this domain leads conclusively to a theory of
+electromagnetic phenomena, of which the law of the constancy of the velocity of
+light in vacuo is a necessary consequence. Prominent theoretical physicists
+were therefore more inclined to reject the principle of relativity, in spite of
+the fact that no empirical data had been found which were contradictory to this
+principle.
+</p>
+
+<p>
+At this juncture the theory of relativity entered the arena. As a result of an
+analysis of the physical conceptions of time and space, it became evident that
+<i>in reality there is not the least incompatibilitiy between the principle of
+relativity and the law of propagation of light</i>, and that by systematically
+holding fast to both these laws a logically rigid theory could be arrived at.
+This theory has been called the <i>special theory of relativity</i> to distinguish it
+from the extended theory, with which we shall deal later. In the following
+pages we shall present the fundamental ideas of the special theory of
+relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap08"></a>VIII.<br/>
+ON THE IDEA OF TIME IN PHYSICS</h3>
+
+<p>
+Lightning has struck the rails on our railway embankment at two places <i>A</i> and <i>B</i>
+far distant from each other. I make the additional assertion that these two
+lightning flashes occurred simultaneously. If I ask you whether there is sense
+in this statement, you will answer my question with a decided “Yes.”
+But if I now approach you with the request to explain to me the sense of the
+statement more precisely, you find after some consideration that the answer to
+this question is not so easy as it appears at first sight.
+</p>
+
+<p>
+After some time perhaps the following answer would occur to you: “The
+significance of the statement is clear in itself and needs no further
+explanation; of course it would require some consideration if I were to be
+commissioned to determine by observations whether in the actual case the two
+events took place simultaneously or not.” I cannot be satisfied with this
+answer for the following reason. Supposing that as a result of ingenious
+considerations an able meteorologist were to discover that the lightning must
+always strike the places <i>A</i> and <i>B</i> simultaneously, then we should be faced with
+the task of testing whether or not this theoretical result is in accordance
+with the reality. We encounter the same difficulty with all physical statements
+in which the conception “simultaneous” plays a part. The concept does
+not exist for the physicist until he has the possibility of discovering whether
+or not it is fulfilled in an actual case. We thus require a definition of
+simultaneity such that this definition supplies us with the method by means of
+which, in the present case, he can decide by experiment whether or not both the
+lightning strokes occurred simultaneously. As long as this requirement is not
+satisfied, I allow myself to be deceived as a physicist (and of course the same
+applies if I am not a physicist), when I imagine that I am able to attach a
+meaning to the statement of simultaneity. (I would ask the reader not to
+proceed farther until he is fully convinced on this point.)
+</p>
+
+<p>
+After thinking the matter over for some time you then offer the following
+suggestion with which to test simultaneity. By measuring along the rails, the
+connecting line <i>AB</i> should be measured up and an observer placed at the
+mid-point M of the distance <i>AB</i>. This observer should be supplied with an
+arrangement (<i>e.g.</i> two mirrors inclined at 90&deg;) which allows him visually to
+observe both places <i>A</i> and <i>B</i> at the same time. If the observer perceives the two
+flashes of lightning at the same time, then they are simultaneous.
+</p>
+
+<p>
+I am very pleased with this suggestion, but for all that I cannot regard the
+matter as quite settled, because I feel constrained to raise the following
+objection:
+“Your definition would certainly be right, if only I knew that the light
+by means of which the observer at <i>M</i> perceives the lightning flashes travels
+along the length <i>A</i> &#8594; <i>M</i> with the same velocity as along the length <i>B</i>
+&#8594; <i>M</i>. But an examination of this supposition would only be possible if we
+already had at our disposal the means of measuring time. It would thus appear
+as though we were moving here in a logical circle.”
+</p>
+
+<p>
+After further consideration you cast a somewhat disdainful glance at me—and
+rightly so—and you declare:
+“I maintain my previous definition nevertheless, because in reality it
+assumes absolutely nothing about light. There is only <i>one</i> demand to be made of
+the definition of simultaneity, namely, that in every real case it must supply
+us with an empirical decision as to whether or not the conception that has to
+be defined is fulfilled. That my definition satisfies this demand is
+indisputable. That light requires the same time to traverse the path <i>A</i> &#8594;
+<i>M</i> as for the path <i>B</i> &#8594; <i>M</i> is in reality neither a <i>supposition nor a
+hypothesis</i> about the physical nature of light, but a <i>stipulation</i> which I can
+make of my own freewill in order to arrive at a definition of
+simultaneity.”
+</p>
+
+<p>
+It is clear that this definition can be used to give an exact meaning not only
+to <i>two</i> events, but to as many events as we care to choose, and
+independently of the positions of the scenes of the events with respect to the
+body of reference<a href="#linknote-7" name="linknoteref-7"
+id="linknoteref-7">[7]</a> (here the railway embankment). We are thus led also
+to a definition of “time” in physics. For this purpose we suppose
+that clocks of identical construction are placed at the points <i>A, B</i> and
+<i>C</i> of the railway line (co-ordinate system) and that they are set in such
+a manner that the positions of their pointers are simultaneously (in the above
+sense) the same. Under these conditions we understand by the “time”
+of an event the reading (position of the hands) of that one of these clocks
+which is in the immediate vicinity (in space) of the event. In this manner a
+time-value is associated with every event which is essentially capable of
+observation.
+</p>
+
+<p>
+<a name="linknote-7" id="linknote-7">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-7"> [7]</a><br/> We suppose further that, when three
+events <i>A, B</i> and <i>C</i> occur in different places in such a manner
+that, if <i>A</i> is simultaneous with <i>B</i>, and <i>B</i> is simultaneous
+with <i>C</i> (simultaneous in the sense of the above definition), then the
+criterion for the simultaneity of the pair of events <i>A, C</i> is also
+satisfied. This assumption is a physical hypothesis about the law of
+propagation of light; it must certainly be fulfilled if we are to maintain the
+law of the constancy of the velocity of light <i>in vacuo</i>.
+</p>
+
+<p>
+This stipulation contains a further physical hypothesis, the validity of which
+will hardly be doubted without empirical evidence to the contrary. It has been
+assumed that all these clocks <i>go at the same rate</i> if they are of identical
+construction. Stated more exactly: When two clocks arranged at rest in
+different places of a reference-body are set in such a manner that a <i>particular</i>
+position of the pointers of the one clock is <i>simultaneous</i> (in the above sense)
+with the <i>same</i> position, of the pointers of the other clock, then identical
+“settings” are always simultaneous (in the sense of the above
+definition).
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap09"></a>IX.<br/>
+THE RELATIVITY OF SIMULTANEITY</h3>
+
+<p>
+Up to now our considerations have been referred to a particular body of
+reference, which we have styled a “railway embankment.” We suppose a
+very long train travelling along the rails with the constant velocity v and in
+the direction indicated in Fig 1. People travelling in this train will with a
+vantage view the train as a rigid reference-body (co-ordinate system); they
+regard all events in reference to the train. Then every event which takes place
+along the line also takes place at a particular point of the train. Also the
+definition of simultaneity can be given relative to the train in exactly the
+same way as with respect to the embankment. As a natural consequence, however,
+the following question arises:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image001.jpg" style="width:100%;" alt="image001" /><br/><br/>
+</div>
+
+<p>
+Are two events (<i>e.g.</i> the two strokes of lightning <i>A</i> and <i>B</i>) which are
+simultaneous <i>with reference to the railway embankment</i> also simultaneous
+<i>relatively to the train?</i> We shall show directly that the answer must be in the
+negative.
+</p>
+
+<p>
+When we say that the lightning strokes <i>A</i> and <i>B</i> are simultaneous with respect to
+be embankment, we mean: the rays of light emitted at the places <i>A</i> and <i>B</i>, where
+the lightning occurs, meet each other at the mid-point <i>M</i> of the length
+<i>A</i> &#8594; <i>B</i> of the embankment. But the events <i>A</i> and <i>B</i> also correspond
+to positions <i>A</i> and <i>B</i> on the train. Let <i>M&#x2032;</i> be the mid-point of the distance <i>A</i>
+&#8594; <i>B</i> on the travelling train. Just when the flashes (as judged from the
+embankment) of lightning occur, this point <i>M&#x2032;</i> naturally coincides with the
+point <i>M</i> but it moves towards the right in the diagram with the velocity v of
+the train. If an observer sitting in the position <i>M&#x2032;</i> in the train did not
+possess this velocity, then he would remain permanently at M, and the light
+rays emitted by the flashes of lightning <i>A</i> and <i>B</i> would reach him
+simultaneously, <i>i.e.</i> they would meet just where he is situated. Now in reality
+(considered with reference to the railway embankment) he is hastening towards
+the beam of light coming from <i>B</i>, whilst he is riding on ahead of the beam of
+light coming from <i>A</i>. Hence the observer will see the beam of light emitted from
+<i>B</i> earlier than he will see that emitted from <i>A</i>. Observers who take the railway
+train as their reference-body must therefore come to the conclusion that the
+lightning flash <i>B</i> took place earlier than the lightning flash <i>A</i>. We thus arrive
+at the important result:
+</p>
+
+<p>
+Events which are simultaneous with reference to the embankment are not
+simultaneous with respect to the train, and <i>vice versa</i> (relativity of
+simultaneity). Every reference-body (co-ordinate system) has its own particular
+time; unless we are told the reference-body to which the statement of time
+refers, there is no meaning in a statement of the time of an event.
+</p>
+
+<p>
+Now before the advent of the theory of relativity it had always tacitly been
+assumed in physics that the statement of time had an absolute significance,
+<i>i.e.</i> that it is independent of the state of motion of the body of reference.
+But we have just seen that this assumption is incompatible with the most
+natural definition of simultaneity; if we discard this assumption, then the
+conflict between the law of the propagation of light <i>in vacuo</i> and the principle
+of relativity (developed in Section VII) disappears.
+</p>
+
+<p>
+We were led to that conflict by the considerations of Section VI, which are now
+no longer tenable. In that section we concluded that the man in the carriage,
+who traverses the distance <i>w per second</i> relative to the carriage, traverses the
+same distance also with respect to the embankment <i>in each second</i> of time. But,
+according to the foregoing considerations, the time required by a particular
+occurrence with respect to the carriage must not be considered equal to the
+duration of the same occurrence as judged from the embankment (as
+reference-body). Hence it cannot be contended that the man in walking travels
+the distance <i>w</i> relative to the railway line in a time which is equal to one
+second as judged from the embankment.
+</p>
+
+<p>
+Moreover, the considerations of Section VI are based on yet a second assumption,
+which, in the light of a strict consideration, appears to be arbitrary,
+although it was always tacitly made even before the introduction of the theory
+of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap10"></a>X.<br/>
+ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE</h3>
+
+<p>
+Let us consider two particular points on the train <a href="#linknote-8" name="linknoteref-8" id="linknoteref-8">[8]</a> travelling
+along the embankment with the velocity <i>v</i>, and inquire as to their distance
+apart. We already know that it is necessary to have a body of reference for the
+measurement of a distance, with respect to which body the distance can be
+measured up. It is the simplest plan to use the train itself as reference-body
+(co-ordinate system). An observer in the train measures the interval by marking
+off his measuring-rod in a straight line (<i>e.g.</i> along the floor of the carriage)
+as many times as is necessary to take him from the one marked point to the
+other. Then the number which tells us how often the rod has to be laid down is
+the required distance.
+</p>
+
+<p>
+<a name="linknote-8" id="linknote-8">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-8"> [8]</a><br/> <i>e.g.</i> the middle of the first and
+of the hundredth carriage.
+</p>
+
+<p>
+It is a different matter when the distance has to be judged from the railway
+line. Here the following method suggests itself. If we call <i>A&#x2032;</i> and <i>B&#x2032;</i> the two
+points on the train whose distance apart is required, then both of these points
+are moving with the velocity v along the embankment. In the first place we
+require to determine the points <i>A</i> and <i>B</i> of the embankment which are just being
+passed by the two points <i>A&#x2032;</i> and <i>B&#x2032;</i> at a particular time t—judged from the
+embankment. These points <i>A</i> and <i>B</i> of the embankment can be determined by
+applying the definition of time given in Section VIII. The distance between these
+points A and B is then measured by repeated application of the measuring-rod
+along the embankment.
+</p>
+
+<p>
+<i>A priori</i> it is by no means certain that this last measurement will supply us
+with the same result as the first. Thus the length of the train as measured
+from the embankment may be different from that obtained by measuring in the
+train itself. This circumstance leads us to a second objection which must be
+raised against the apparently obvious consideration of Section VI. Namely, if
+the man in the carriage covers the distance <i>w</i> in a unit of time—<i>measured from
+the train</i>,—then this distance—<i>as measured from the embankment</i> is not
+necessarily also equal to <i>w</i>.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap11"></a>XI.<br/>
+THE LORENTZ TRANSFORMATION</h3>
+
+<p>
+The results of the last three sections show that the apparent incompatibility
+of the law of propagation of light with the principle of relativity (Section VII)
+has been derived by means of a consideration which borrowed two unjustifiable
+hypotheses from classical mechanics; these are as follows:
+</p>
+
+<p class="letter">
+(1) The time-interval (time) between two events is independent of the condition
+of motion of the body of reference.
+</p>
+
+<p class="letter">
+(2) The space-interval (distance) between two points of a rigid body is
+independent of the condition of motion of the body of reference.
+</p>
+
+<p>
+If we drop these hypotheses, then the dilemma of Section VII disappears, because
+the theorem of the addition of velocities derived in Section VI becomes invalid.
+The possibility presents itself that the law of the propagation of light <i>in
+vacuo</i> may be compatible with the principle of relativity, and the question
+arises: How have we to modify the considerations of Section VI in order to
+remove the apparent disagreement between these two fundamental results of
+experience? This question leads to a general one. In the discussion of Section
+VI we have to do with places and times relative both to the train and to the
+embankment. How are we to find the place and time of an event in relation to
+the train, when we know the place and time of the event with respect to the
+railway embankment? Is there a thinkable answer to this question of such a
+nature that the law of transmission of light <i>in vacuo</i> does not contradict the
+principle of relativity? In other words: Can we conceive of a relation between
+place and time of the individual events relative to both reference-bodies, such
+that every ray of light possesses the velocity of transmission <i>c</i> relative to
+the embankment and relative to the train? This question leads to a quite
+definite positive answer, and to a perfectly definite transformation law for
+the space-time magnitudes of an event when changing over from one body of
+reference to another.
+</p>
+
+<p>
+Before we deal with this, we shall introduce the following incidental
+consideration. Up to the present we have only considered events taking place
+along the embankment, which had mathematically to assume the function of a
+straight line. In the manner indicated in Section II we can imagine this
+reference-body supplemented laterally and in a vertical direction by means of a
+framework of rods, so that an event which takes place anywhere can be localised
+with reference to this framework.
+Similarly, we can imagine the train travelling with the velocity <i>v</i> to be
+continued across the whole of space, so that every event, no matter how far off
+it may be, could also be localised with respect to the second framework.
+Without committing any fundamental error, we can disregard the fact that in
+reality these frameworks would continually interfere with each other, owing to
+the impenetrability of solid bodies. In every such framework we imagine three
+surfaces perpendicular to each other marked out, and designated as
+“co-ordinate planes” (“co-ordinate system”). A co-ordinate
+system <i>K</i> then corresponds to the embankment, and a co-ordinate system <i>K&#x2032;</i> to the
+train. An event, wherever it may have taken place, would be fixed in space with
+respect to <i>K</i> by the three perpendiculars <i>x, y, z</i> on the co-ordinate planes, and
+with regard to time by a time value <i>t</i>. Relative to <i>K&#x2032;, the same event</i> would be
+fixed in respect of space and time by corresponding values <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>,
+which of course are not identical with <i>x, y, z, t</i>. It has already been set
+forth in detail how these magnitudes are to be regarded as results of physical
+measurements.
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image002.jpg" style="width:100%;" alt="image002" /><br/><br/>
+</div>
+
+<p>
+Obviously our problem can be exactly formulated in the following manner. What
+are the values <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>, of an event with respect to <i>K&#x2032;</i>, when the
+magnitudes <i>x, y, z, t</i>, of the same event with respect to <i>K</i> are given? The
+relations must be so chosen that the law of the transmission of light in vacuo
+is satisfied for one and the same ray of light (and of course for every ray)
+with respect to <i>K</i> and <i>K&#x2032;</i>. For the relative orientation in space of the
+co-ordinate systems indicated in the diagram (Fig. 2), this problem is solved
+by means of the equations:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image003.jpg" style="width:100%;" alt="image003" /><br/><br/>
+</div>
+
+<p class="center">
+<i>y&#x2032;</i> = <i>y</i>
+</p>
+<p class="center">
+<i>z&#x2032;</i> = <i>z</i>
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image004.jpg" style="width:100%;" alt="image004" /><br/><br/>
+</div>
+
+<p>
+This system of equations is known as the “Lorentz
+transformation.”<a href="#linknote-9" name="linknoteref-9" id="linknoteref-9">[9]</a>
+</p>
+
+<p>
+<a name="linknote-9" id="linknote-9">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-9"> [9]</a><br/> A simple derivation of the Lorentz
+transformation is given in Appendix I.
+</p>
+
+<p>
+If in place of the law of transmission of light we had taken as our basis the
+tacit assumptions of the older mechanics as to the absolute character of times
+and lengths, then instead of the above we should have obtained the following
+equations:
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>x</i> – <i>vt</i>
+</p>
+
+<p class="center">
+<i>y&#x2032;</i> = <i>y</i>
+</p>
+
+<p class="center">
+<i>z&#x2032;</i> = <i>z</i>
+</p>
+
+<p class="center">
+<i>t&#x2032;</i> = <i>t</i>
+</p>
+
+<p>
+This system of equations is often termed the “Galilei
+transformation.” The Galilei transformation can be obtained from the
+Lorentz transformation by substituting an infinitely large value for the
+velocity of light <i>c</i> in the latter transformation.
+</p>
+
+<p>
+Aided by the following illustration, we can readily see that, in accordance
+with the Lorentz transformation, the law of the transmission of light <i>in vacuo</i>
+is satisfied both for the reference-body <i>K</i> and for the reference-body <i>K&#x2032;</i>. A
+light-signal is sent along the positive <i>x</i>-axis, and this light-stimulus
+advances in accordance with the equation
+</p>
+
+<p class="center">
+<i>x</i> = <i>ct</i>,
+</p>
+
+<p class="noindent">
+<i>i.e.</i> with the velocity <i>c</i>. According to the equations of the Lorentz
+transformation, this simple relation between <i>x</i> and <i>t</i> involves a relation
+between <i>x&#x2032;</i> and <i>t&#x2032;</i>. In point of fact, if we substitute for <i>x</i> the value <i>ct</i> in the
+first and fourth equations of the Lorentz transformation, we obtain:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image005.jpg" style="width:100%;" alt="image005" /><br/><br/>
+</div>
+
+<p class="noindent">
+from which, by division, the expression
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>ct&#x2032;</i>
+</p>
+
+<p class="noindent">
+immediately follows. If referred to the system <i>K&#x2032;</i>, the propagation of light
+takes place according to this equation. We thus see that the velocity of
+transmission relative to the reference-body <i>K&#x2032;</i> is also equal to <i>c</i>. The same
+result is obtained for rays of light advancing in any other direction
+whatsoever. Of cause this is not surprising, since the equations of the Lorentz
+transformation were derived conformably to this point of view.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap12"></a>XII.<br/>
+THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION</h3>
+
+<p>
+Place a metre-rod in the <i>x&#x2032;</i>-axis of <i>K&#x2032;</i> in such a manner that one end (the
+beginning) coincides with the point <i>x&#x2032;</i> = 0 whilst the other end (the end of the
+rod) coincides with the point <i>x&#x2032;</i> = 1. What is the length of the metre-rod
+relatively to the system <i>K</i>? In order to learn this, we need only ask where the
+beginning of the rod and the end of the rod lie with respect to <i>K</i> at a
+particular time <i>t</i> of the system <i>K</i>. By means of the first equation of the
+Lorentz transformation the values of these two points at the time <i>t</i> = 0 can be
+shown to be
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image006.jpg" style="width:100%;" alt="image006" /><br/><br/>
+</div>
+
+<p class="noindent">
+the distance between the points being
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image007.jpg" style="width:100%;" alt="image007" /><br/><br/>
+</div>
+
+<p class="noindent">
+But the metre-rod is moving with the velocity <i>v</i> relative to <i>K</i>. It therefore
+follows that the length of a rigid metre-rod moving in the direction of its
+length with a velocity <i>v</i> is
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image008.jpg" style="width:100%;" alt="image008" /><br/><br/>
+</div>
+
+<p class="noindent">
+of a metre. The rigid rod is thus shorter when in motion than when at rest, and
+the more quickly it is moving, the shorter is the rod. For the velocity <i>v</i> = <i>c</i> we
+should have
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image009.jpg" style="width:100%;" alt="image009" /><br/><br/>
+</div>
+
+<p class="noindent">
+and for still greater velocities the square-root becomes imaginary. From this
+we conclude that in the theory of relativity the velocity <i>c</i> plays the
+part of a limiting velocity, which can neither be reached nor exceeded by any
+real body.
+</p>
+
+<p>
+Of course this feature of the velocity <i>c</i> as a limiting velocity also clearly
+follows from the equations of the Lorentz transformation, for these became
+meaningless if we choose values of <i>v</i> greater than <i>c</i>.
+</p>
+
+<p>
+If, on the contrary, we had considered a metre-rod at rest in the <i>x</i>-axis with
+respect to <i>K</i>, then we should have found that the length of the rod as judged
+from <i>K&#x2032;</i> would have been
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image010.jpg" style="width:100%;" alt="image010" /><br/><br/>
+</div>
+
+<p class="noindent">
+this is quite in accordance with the principle of relativity which forms the
+basis of our considerations.
+</p>
+
+<p>
+<i>A priori</i> it is quite clear that we must be able to learn something about the
+physical behaviour of measuring-rods and clocks from the equations of
+transformation, for the magnitudes <i>z, y, x, t</i>, are nothing more nor less than
+the results of measurements obtainable by means of measuring-rods and clocks.
+If we had based our considerations on the Galileian transformation we should
+not have obtained a contraction of the rod as a consequence of its motion.
+</p>
+
+<p>
+Let us now consider a seconds-clock which is permanently situated at the origin
+(<i>x&#x2032;</i> = 0) of <i>K&#x2032;</i>. <i>t&#x2032;</i> = 0 and <i>t&#x2032;</i> = 1 are two successive ticks of this clock. The first
+and fourth equations of the Lorentz transformation give for these two ticks:
+</p>
+
+<p>
+<i>t</i> = 0
+</p>
+
+<p class="noindent">
+and
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image011.jpg" style="width:100%;" alt="image011" /><br/><br/>
+</div>
+
+<p>
+As judged from <i>K</i>, the clock is moving with the velocity <i>v</i>; as judged from this
+reference-body, the time which elapses between two strokes of the clock is not
+one second, but
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image012.jpg" style="width:100%;" alt="image012" /><br/><br/>
+</div>
+
+<p class="noindent">
+seconds, <i>i.e.</i> a somewhat larger time. As a consequence of its motion the clock
+goes more slowly than when at rest. Here also the velocity <i>c</i> plays the part of
+an unattainable limiting velocity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap13"></a>XIII.<br/>
+THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU</h3>
+
+<p>
+Now in practice we can move clocks and measuring-rods only with velocities that
+are small compared with the velocity of light; hence we shall hardly be able to
+compare the results of the previous section directly with the reality. But, on
+the other hand, these results must strike you as being very singular, and for
+that reason I shall now draw another conclusion from the theory, one which can
+easily be derived from the foregoing considerations, and which has been most
+elegantly confirmed by experiment.
+</p>
+
+<p>
+In Section VI we derived the theorem of the addition of velocities in one
+direction in the form which also results from the hypotheses of classical
+mechanics. This theorem can also be deduced readily from the Galilei
+transformation (Section XI). In place of the man walking inside the carriage,
+we introduce a point moving relatively to the co-ordinate system <i>K&#x2032;</i> in
+accordance with the equation
+</p>
+
+<p>
+<i>x&#x2032;</i> = <i>wt&#x2032;</i>
+</p>
+
+<p class="noindent">
+By means of the first and fourth equations of the Galilei transformation we can
+express <i>x&#x2032;</i> and <i>t&#x2032;</i> in terms of <i>x</i> and <i>t</i>, and we then obtain
+</p>
+
+<p>
+<i>x</i> = (<i>v</i> + <i>w</i>)<i>t</i>
+</p>
+
+<p class="noindent">
+This equation expresses nothing else than the law of motion of the point with
+reference to the system <i>K</i> (of the man with reference to the embankment). We
+denote this velocity by the symbol <i>W</i>, and we then obtain, as in Section VI,
+</p>
+
+<p>
+<i>W</i> = <i>v</i> + <i>w</i> . . . . . . . (A).
+</p>
+
+<p>
+But we can carry out this consideration just as well on the basis of the theory
+of relativity. In the equation
+</p>
+
+<p>
+<i>x&#x2032;</i> = <i>wt&#x2032;</i>
+</p>
+
+<p>
+we must then express <i>x&#x2032;</i> and <i>t&#x2032;</i> in terms of <i>x</i> and <i>t</i>, making use of the first and
+fourth equations of the <i>Lorentz transformation</i>. Instead of the equation (A) we
+then obtain the equation
+</p>
+
+<div class="fig" style="width:50%;">
+<img src="images/image013.jpg" style="width:100%;" alt="image013" /><br/><br/>
+</div>
+
+<p class="noindent">
+which corresponds to the theorem of addition for velocities in one direction
+according to the theory of relativity. The question now arises as to which of
+these two theorems is the better in accord with experience. On this point we
+are enlightened by a most important experiment which the brilliant physicist
+Fizeau performed more than half a century ago, and which has been repeated
+since then by some of the best experimental physicists, so that there can be no
+doubt about its result. The experiment is concerned with the following
+question. Light travels in a motionless liquid with a particular velocity
+<i>w</i>. How quickly does it travel in the direction of the arrow in the tube
+<i>T</i> (see the accompanying diagram, Fig. 3) when the liquid above mentioned
+is flowing through the tube with a velocity <i>v</i>?
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image014.jpg" style="width:100%;" alt="image014" /><br/><br/>
+</div>
+
+<p>
+In accordance with the principle of relativity we shall certainly have to take
+for granted that the propagation of light always takes place with the same
+velocity <i>w with respect to the liquid</i>, whether the latter is in motion with
+reference to other bodies or not. The velocity of light relative to the liquid
+and the velocity of the latter relative to the tube are thus known, and we
+require the velocity of light relative to the tube.
+</p>
+
+<p>
+It is clear that we have the problem of Section VI again before us. The tube
+plays the part of the railway embankment or of the co-ordinate system <i>K</i>,
+the liquid plays the part of the carriage or of the co-ordinate system
+<i>K&#x2032;</i>, and finally, the light plays the part of the man walking
+along the carriage, or of the moving point in the present section. If we denote
+the velocity of the light relative to the tube by <i>W</i>, then this is given
+by the equation (A) or (B), according as the Galilei transformation or the
+Lorentz transformation corresponds to the facts. Experiment<a
+href="#linknote-10" name="linknoteref-10" id="linknoteref-10">[10]</a> decides
+in favour of equation (B) derived from the theory of relativity, and the
+agreement is, indeed, very exact. According to recent and most excellent
+measurements by Zeeman, the influence of the velocity of flow <i>v</i> on the
+propagation of light is represented by formula (B) to within one per cent.
+</p>
+
+<p>
+<a name="linknote-10" id="linknote-10">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-10"> [10]</a><br/> Fizeau found
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image015.jpg" style="width:100%;" alt="image015" /><br/><br/>
+</div>
+
+<p class="footnote">
+where
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image016.jpg" style="width:100%;" alt="image016" /><br/><br/>
+</div>
+
+<p class="footnote">
+is the index of refraction of the liquid. On the other hand, owing to the
+smallness of
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image017.jpg" style="width:100%;" alt="image017" /><br/><br/>
+</div>
+
+<p class="footnote">
+as compared with 1, we can replace (B) in the first place by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image018.jpg" style="width:100%;" alt="image018" /><br/><br/>
+</div>
+
+<p class="footnote">
+or to the same order of approximation by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image019.jpg" style="width:100%;" alt="image019" /><br/><br/>
+</div>
+
+<p class="footnote">
+which agrees with Fizeau’s result.
+</p>
+
+<p>
+Nevertheless we must now draw attention to the fact that a theory of this
+phenomenon was given by H. A. Lorentz long before the statement of the theory
+of relativity. This theory was of a purely electrodynamical nature, and was
+obtained by the use of particular hypotheses as to the electromagnetic
+structure of matter. This circumstance, however, does not in the least diminish
+the conclusiveness of the experiment as a crucial test in favour of the theory
+of relativity, for the electrodynamics of Maxwell-Lorentz, on which the
+original theory was based, in no way opposes the theory of relativity. Rather
+has the latter been developed trom electrodynamics as an astoundingly simple
+combination and generalisation of the hypotheses, formerly independent of each
+other, on which electrodynamics was built.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap14"></a>XIV.<br/>
+THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY</h3>
+
+<p>
+Our train of thought in the foregoing pages can be epitomised in the following
+manner. Experience has led to the conviction that, on the one hand, the
+principle of relativity holds true and that on the other hand the velocity of
+transmission of light <i>in vacuo</i> has to be considered equal to a constant <i>c</i>. By
+uniting these two postulates we obtained the law of transformation for the
+rectangular co-ordinates <i>x, y, z</i> and the time <i>t</i> of the events which constitute
+the processes of nature. In this connection we did not obtain the Galilei
+transformation, but, differing from classical mechanics, the <i>Lorentz
+transformation</i>.
+</p>
+
+<p>
+The law of transmission of light, the acceptance of which is justified by our
+actual knowledge, played an important part in this process of thought. Once in
+possession of the Lorentz transformation, however, we can combine this with the
+principle of relativity, and sum up the theory thus:
+</p>
+
+<p>
+Every general law of nature must be so constituted that it is transformed into
+a law of exactly the same form when, instead of the space-time variables <i>x, y,
+z, t</i> of the original coordinate system <i>K</i>, we introduce new space-time variables
+<i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i> of a co-ordinate system <i>K&#x2032;</i>. In this connection the relation
+between the ordinary and the accented magnitudes is given by the Lorentz
+transformation. Or in brief: General laws of nature are co-variant with
+respect to Lorentz transformations.
+</p>
+
+<p>
+This is a definite mathematical condition that the theory of relativity demands
+of a natural law, and in virtue of this, the theory becomes a valuable
+heuristic aid in the search for general laws of nature. If a general law of
+nature were to be found which did not satisfy this condition, then at least one
+of the two fundamental assumptions of the theory would have been disproved. Let
+us now examine what general results the latter theory has hitherto evinced.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap15"></a>XV.<br/>
+GENERAL RESULTS OF THE THEORY</h3>
+
+<p>
+It is clear from our previous considerations that the (special) theory of
+relativity has grown out of electrodynamics and optics. In these fields it has
+not appreciably altered the predictions of theory, but it has considerably
+simplified the theoretical structure, <i>i.e.</i> the derivation of laws, and—what is
+incomparably more important—it has considerably reduced the number of
+independent hypotheses forming the basis of theory. The special theory of
+relativity has rendered the Maxwell-Lorentz theory so plausible, that the
+latter would have been generally accepted by physicists even if experiment had
+decided less unequivocally in its favour.
+</p>
+
+<p>
+Classical mechanics required to be modified before it could come into line with
+the demands of the special theory of relativity. For the main part, however,
+this modification affects only the laws for rapid motions, in which the
+velocities of matter <i>v</i> are not very small as compared with the velocity of
+light. We have experience of such rapid motions only in the case of electrons
+and ions; for other motions the variations from the laws of classical mechanics
+are too small to make themselves evident in practice. We shall not consider the
+motion of stars until we come to speak of the general theory of relativity. In
+accordance with the theory of relativity the kinetic energy of a material point
+of mass <i>m</i> is no longer given by the well-known expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image020.jpg" style="width:100%;" alt="image020" /><br/><br/>
+</div>
+
+<p class="noindent">
+but by the expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image021.jpg" style="width:100%;" alt="image021" /><br/><br/>
+</div>
+
+<p class="noindent">
+This expression approaches infinity as the velocity <i>v</i> approaches the velocity
+of light <i>c</i>. The velocity must therefore always remain less than <i>c</i>, however
+great may be the energies used to produce the acceleration. If we develop the
+expression for the kinetic energy in the form of a series, we obtain
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image022.jpg" style="width:100%;" alt="image022" /><br/><br/>
+</div>
+
+<p>
+When
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image023.jpg" style="width:100%;" alt="image023" /><br/><br/>
+</div>
+
+<p class="noindent">
+is small compared with unity, the third of these terms is always small in
+comparison with the second, which last is alone considered in classical
+mechanics. The first term <i>mc</i><sup>2</sup> does not contain the velocity, and
+requires no consideration if we are only dealing with the question as to how
+the energy of a point-mass; depends on the velocity. We shall speak of its
+essential significance later.
+</p>
+
+<p>
+The most important result of a general character to which the special theory of
+relativity has led is concerned with the conception of mass. Before the advent
+of relativity, physics recognised two conservation laws of fundamental
+importance, namely, the law of the conservation of energy and the law of the
+conservation of mass these two fundamental laws appeared to be quite
+independent of each other. By means of the theory of relativity they have been
+united into one law. We shall now briefly consider how this unification came
+about, and what meaning is to be attached to it.
+</p>
+
+<p>
+The principle of relativity requires that the law of the conservation of energy
+should hold not only with reference to a co-ordinate system <i>K</i>, but also with
+respect to every co-ordinate system <i>K&#x2032;</i> which is in a state of uniform motion of
+translation relative to <i>K</i>, or, briefly, relative to every “Galileian”
+system of co-ordinates. In contrast to classical mechanics; the Lorentz
+transformation is the deciding factor in the transition from one such system to
+another.
+</p>
+
+<p>
+By means of comparatively simple considerations we are led to draw the
+following conclusion from these premises, in conjunction with the fundamental
+equations of the electrodynamics of Maxwell: A body moving with the velocity <i>v</i>,
+which absorbs<a href="#linknote-11" name="linknoteref-11" id="linknoteref-11">[11]</a> an amount of energy <i>E</i><sub>0</sub> in the form of
+radiation without suffering an alteration in velocity in the process, has, as a
+consequence, its energy increased by an amount
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image024.jpg" style="width:100%;" alt="image024" /><br/><br/>
+</div>
+
+<p>
+<a name="linknote-11" id="linknote-11">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-11"> [11]</a><br/> <i>E</i><sub>0</sub> is the energy
+taken up, as judged from a co-ordinate system moving with the body.
+</p>
+
+<p>
+In consideration of the expression given above for the kinetic energy of the
+body, the required energy of the body comes out to be
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image025.jpg" style="width:100%;" alt="image025" /><br/><br/>
+</div>
+
+<p>
+Thus the body has the same energy as a body of mass
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image026.jpg" style="width:100%;" alt="image026" /><br/><br/>
+</div>
+
+<p class="noindent">
+moving with the velocity <i>v</i>. Hence we can say: If a body takes up an amount of
+energy <i>E</i><sub>0</sub>, then its inertial mass increases by an amount
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image027.jpg" style="width:100%;" alt="image027" /><br/><br/>
+</div>
+
+<p class="noindent">
+the inertial mass of a body is not a constant but varies according to the
+change in the energy of the body. The inertial mass of a system of bodies can
+even be regarded as a measure of its energy. The law of the conservation of the
+mass of a system becomes identical with the law of the conservation of energy,
+and is only valid provided that the system neither takes up nor sends out
+energy. Writing the expression for the energy in the form
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image028.jpg" style="width:100%;" alt="image028" /><br/><br/>
+</div>
+
+<p class="noindent">
+we see that the term <i>mc</i><sup>2</sup>, which has hitherto attracted our
+attention, is nothing else than the energy possessed by the body<a href="#linknote-12" name="linknoteref-12" id="linknoteref-12">[12]</a>
+before it absorbed the energy <i>E</i><sub>0</sub>.
+</p>
+
+<p>
+<a name="linknote-12" id="linknote-12">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-12"> [12]</a><br/> As judged from a co-ordinate system
+moving with the body.
+</p>
+
+<p>
+A direct comparison of this relation with experiment is not possible at the
+present time (1920; see<a href="#linknote-Note" name="linknoteref-Note" id="linknoteref-Note">[Note]</a>, p. 48), owing to the fact that the
+changes in energy <i>E</i><sub>0</sub> to which we can subject a system are not large
+enough to make themselves perceptible as a change in the inertial mass of the
+system.
+</p>
+
+<div class="fig" style="width:15%;">
+<img src="images/image027.jpg" style="width:100%;" alt="image027" /><br/><br/>
+</div>
+
+<p class="noindent">
+is too small in comparison with the mass <i>m</i>, which was present before the
+alteration of the energy. It is owing to this circumstance that classical
+mechanics was able to establish successfully the conservation of mass as a law
+of independent validity.
+</p>
+
+<p>
+<a name="linknote-Note" id="linknote-Note">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-Note"> [Note]</a><br/> The equation E = mc<sup>2</sup>
+has been thoroughly proved time and again since this time.
+</p>
+
+<p>
+Let me add a final remark of a fundamental nature. The success of the
+Faraday-Maxwell interpretation of electromagnetic action at a distance resulted
+in physicists becoming convinced that there are no such things as instantaneous
+actions at a distance (not involving an intermediary medium) of the type of
+Newton’s law of gravitation.
+</p>
+
+<p>
+According to the theory of relativity, action at a distance with the velocity
+of light always takes the place of instantaneous action at a distance or of
+action at a distance with an infinite velocity of transmission. This is
+connected with the fact that the velocity <i>c</i> plays a fundamental role in this
+theory. In Part II we shall see in what way this result becomes modified in the
+general theory of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap16"></a>XVI.<br/>
+EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY</h3>
+
+<p>
+To what extent is the special theory of relativity supported by experience?
+This question is not easily answered for the reason already mentioned in
+connection with the fundamental experiment of Fizeau. The special theory of
+relativity has crystallised out from the Maxwell-Lorentz theory of
+electromagnetic phenomena. Thus all facts of experience which support the
+electromagnetic theory also support the theory of relativity. As being of
+particular importance, I mention here the fact that the theory of relativity
+enables us to predict the effects produced on the light reaching us from the
+fixed stars. These results are obtained in an exceedingly simple manner, and
+the effects indicated, which are due to the relative motion of the earth with
+reference to those fixed stars are found to be in accord with experience. We
+refer to the yearly movement of the apparent position of the fixed stars
+resulting from the motion of the earth round the sun (aberration), and to the
+influence of the radial components of the relative motions of the fixed stars
+with respect to the earth on the colour of the light reaching us from them. The
+latter effect manifests itself in a slight displacement of the spectral lines
+of the light transmitted to us from a fixed star, as compared with the position
+of the same spectral lines when they are produced by a terrestrial source of
+light (Doppler principle). The experimental arguments in favour of the
+Maxwell-Lorentz theory, which are at the same time arguments in favour of the
+theory of relativity, are too numerous to be set forth here. In reality they
+limit the theoretical possibilities to such an extent, that no other theory
+than that of Maxwell and Lorentz has been able to hold its own when tested by
+experience.
+</p>
+
+<p>
+But there are two classes of experimental facts hitherto obtained which can be
+represented in the Maxwell-Lorentz theory only by the introduction of an
+auxiliary hypothesis, which in itself—<i>i.e.</i> without making use of the theory of
+relativity—appears extraneous.
+</p>
+
+<p>
+It is known that cathode rays and the so-called &#946;-rays emitted by
+radioactive substances consist of negatively electrified particles (electrons)
+of very small inertia and large velocity. By examining the deflection of these
+rays under the influence of electric and magnetic fields, we can study the law
+of motion of these particles very exactly.
+</p>
+
+<p>
+In the theoretical treatment of these electrons, we are faced with the
+difficulty that electrodynamic theory of itself is unable to give an account of
+their nature. For since electrical masses of one sign repel each other, the
+negative electrical masses constituting the electron would necessarily be
+scattered under the influence of their mutual repulsions, unless there are
+forces of another kind operating between them, the nature of which has hitherto
+remained obscure to us.<a href="#linknote-13" name="linknoteref-13" id="linknoteref-13">[13]</a> If we now assume that the relative
+distances between the electrical masses constituting the electron remain
+unchanged during the motion of the electron (rigid connection in the sense of
+classical mechanics), we arrive at a law of motion of the electron which does
+not agree with experience. Guided by purely formal points of view, H. A.
+Lorentz was the first to introduce the hypothesis that the form of the electron
+experiences a contraction in the direction of motion in consequence of that
+motion. the contracted length being proportional to the expression
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image029.jpg" style="width:100%;" alt="image029" /><br/><br/>
+</div>
+
+<p class="noindent">
+This, hypothesis, which is not justifiable by any electrodynamical facts,
+supplies us then with that particular law of motion which has been confirmed
+with great precision in recent years.
+</p>
+
+<p>
+<a name="linknote-13" id="linknote-13">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-13"> [13]</a><br/> The general theory of relativity
+renders it likely that the electrical masses of an electron are held together
+by gravitational forces.
+</p>
+
+<p>
+The theory of relativity leads to the same law of motion, without requiring any
+special hypothesis whatsoever as to the structure and the behaviour of the
+electron. We arrived at a similar conclusion in Section XIII in connection with
+the experiment of Fizeau, the result of which is foretold by the theory of
+relativity without the necessity of drawing on hypotheses as to the physical
+nature of the liquid.
+</p>
+
+<p>
+The second class of facts to which we have alluded has reference to the
+question whether or not the motion of the earth in space can be made
+perceptible in terrestrial experiments. We have already remarked in Section V
+that all attempts of this nature led to a negative result. Before the theory of
+relativity was put forward, it was difficult to become reconciled to this
+negative result, for reasons now to be discussed. The inherited prejudices
+about time and space did not allow any doubt to arise as to the prime
+importance of the Galileian transformation for changing over from one body of
+reference to another. Now assuming that the Maxwell-Lorentz equations hold for
+a reference-body <i>K</i>, we then find that they do not hold for a reference-body <i>K&#x2032;</i>
+moving uniformly with respect to <i>K</i>, if we assume that the relations of the
+Galileian transformation exist between the co-ordinates of <i>K</i> and <i>K&#x2032;</i>. It thus
+appears that, of all Galileian co-ordinate systems, one (<i>K</i>) corresponding to a
+particular state of motion is physically unique. This result was interpreted
+physically by regarding <i>K</i> as at rest with respect to a hypothetical æther of
+space. On the other hand, all coordinate systems <i>K&#x2032;</i> moving relatively to <i>K</i> were
+to be regarded as in motion with respect to the æther. To this motion of <i>K&#x2032;</i>
+against the æther (“æther-drift” relative to <i>K&#x2032;</i>) were attributed the
+more complicated laws which were supposed to hold relative to <i>K&#x2032;</i>. Strictly
+speaking, such an æther-drift ought also to be assumed relative to the earth,
+and for a long time the efforts of physicists were devoted to attempts to
+detect the existence of an æther-drift at the earth’s surface.
+</p>
+
+<p>
+In one of the most notable of these attempts Michelson devised a method which
+appears as though it must be decisive. Imagine two mirrors so arranged on a
+rigid body that the reflecting surfaces face each other. A ray of light
+requires a perfectly definite time <i>T</i> to pass from one mirror to the other and
+back again, if the whole system be at rest with respect to the æther. It is
+found by calculation, however, that a slightly different time <i>T&#x2032;</i> is required
+for this process, if the body, together with the mirrors, be moving relatively
+to the æther. And yet another point: it is shown by calculation that for a
+given velocity <i>v</i> with reference to the æther, this time <i>T&#x2032;</i> is different when
+the body is moving perpendicularly to the planes of the mirrors from that
+resulting when the motion is parallel to these planes. Although the estimated
+difference between these two times is exceedingly small, Michelson and Morley
+performed an experiment involving interference in which this difference should
+have been clearly detectable. But the experiment gave a negative result—a fact
+very perplexing to physicists. Lorentz and FitzGerald rescued the theory from
+this difficulty by assuming that the motion of the body relative to the æther
+produces a contraction of the body in the direction of motion, the amount of
+contraction being just sufficient to compensate for the difference in time
+mentioned above. Comparison with the discussion in Section XII shows that also
+from the standpoint of the theory of relativity this solution of the difficulty
+was the right one. But on the basis of the theory of relativity the method of
+interpretation is incomparably more satisfactory. According to this theory
+there is no such thing as a “specially favoured” (unique) co-ordinate
+system to occasion the introduction of the æther-idea, and hence there can be
+no æther-drift, nor any experiment with which to demonstrate it. Here the
+contraction of moving bodies follows from the two fundamental principles of the
+theory, without the introduction of particular hypotheses; and as the prime
+factor involved in this contraction we find, not the motion in itself, to which
+we cannot attach any meaning, but the motion with respect to the body of
+reference chosen in the particular case in point. Thus for a co-ordinate system
+moving with the earth the mirror system of Michelson and Morley is not
+shortened, but it <i>is</i> shortened for a co-ordinate system which is at rest
+relatively to the sun.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap17"></a>XVII.<br/>
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE</h3>
+
+<p>
+The non-mathematician is seized by a mysterious shuddering when he hears of
+“four-dimensional” things, by a feeling not unlike that awakened by
+thoughts of the occult. And yet there is no more common-place statement than
+that the world in which we live is a four-dimensional space-time continuum.
+</p>
+
+<p>
+Space is a three-dimensional continuum. By this we mean that it is possible to
+describe the position of a point (at rest) by means of three numbers
+(co-ordinates) <i>x, y, z</i>, and that there is an indefinite number of points in the
+neighbourhood of this one, the position of which can be described by
+co-ordinates such as <i>x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub></i>, which may be
+as near as we choose to the respective values of the co-ordinates <i>x, y, z</i>, of
+the first point. In virtue of the latter property we speak of a
+“continuum,” and owing to the fact that there are three co-ordinates
+we speak of it as being “three-dimensional.”
+</p>
+
+<p>
+Similarly, the world of physical phenomena which was briefly called
+“world” by Minkowski is naturally four dimensional in the space-time
+sense. For it is composed of individual events, each of which is described by
+four numbers, namely, three space co-ordinates <i>x, y, z</i>, and a time co-ordinate,
+the time value <i>t</i>. The “world” is in this sense
+also a continuum; for to every event there are as many “neighbouring”
+events (realised or at least thinkable) as we care to choose, the co-ordinates
+<i>x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>, t<sub>1</sub></i> of which differ by
+an indefinitely small amount from those of the event <i>x, y, z, t</i> originally
+considered. That we have not been accustomed to regard the world in this sense
+as a four-dimensional continuum is due to the fact that in physics, before the
+advent of the theory of relativity, time played a different and more
+independent rôle, as compared with the space coordinates. It is for this reason
+that we have been in the habit of treating time as an independent continuum. As
+a matter of fact, according to classical mechanics, time is absolute, <i>i.e.</i> it
+is independent of the position and the condition of motion of the system of
+co-ordinates. We see this expressed in the last equation of the Galileian
+transformation (<i>t&#x2032;</i> = <i>t</i>).
+</p>
+
+<p>
+The four-dimensional mode of consideration of the “world” is natural
+on the theory of relativity, since according to this theory time is robbed of
+its independence. This is shown by the fourth equation of the Lorentz
+transformation:
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image030.jpg" style="width:100%;" alt="image030" /><br/><br/>
+</div>
+
+<p class="noindent">
+Moreover, according to this equation the time difference &#916;<i>t&#x2032;</i> of two events
+with respect to <i>K&#x2032;</i> does not in general vanish, even when the time difference
+&#916;<i>t</i> of the same events with reference to <i>K</i> vanishes. Pure
+“space-distance” of two events with respect to <i>K</i> results in
+“time-distance ” of the same events with respect to <i>K</i>. But the
+discovery of Minkowski, which was of importance for the formal development of
+the theory of relativity, does not lie here. It is to be found rather in the
+fact of his recognition that the four-dimensional space-time continuum of the
+theory of relativity, in its most essential formal properties, shows a
+pronounced relationship to the three-dimensional continuum of Euclidean
+geometrical space.<a href="#linknote-14" name="linknoteref-14" id="linknoteref-14">[14]</a> In order to give due prominence to this
+relationship, however, we must replace the usual time co-ordinate t by an
+imaginary magnitude
+</p>
+
+<div class="fig" style="width:10%;">
+<img src="images/image031.jpg" style="width:100%;" alt="image031" /><br/><br/>
+</div>
+
+<p class="noindent">
+proportional to it. Under these conditions, the natural laws satisfying the
+demands of the (special) theory of relativity assume mathematical forms, in
+which the time co-ordinate plays exactly the same role as the three space
+co-ordinates. Formally, these four co-ordinates correspond exactly to the
+three space co-ordinates in Euclidean geometry. It must be clear even to the
+non-mathematician that, as a consequence of this purely formal addition to our
+knowledge, the theory perforce gained clearness in no mean measure.
+</p>
+
+<p>
+<a name="linknote-14" id="linknote-14">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-14"> [14]</a><br/> Cf. the somewhat more detailed
+discussion in Appendix II.
+</p>
+
+<p>
+These inadequate remarks can give the reader only a vague notion of the
+important idea contributed by Minkowski. Without it the general theory of
+relativity, of which the fundamental ideas are developed in the following
+pages, would perhaps have got no farther than its long clothes. Minkowski’s
+work is doubtless difficult of access to anyone inexperienced in mathematics,
+but since it is not necessary to have a very exact grasp of this work in order
+to understand the fundamental ideas of either the special or the general theory
+of relativity, I shall leave it here at present, and revert to it only towards
+the end of Part II.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part02"></a>PART II: THE GENERAL THEORY OF RELATIVITY</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap18"></a>XVIII.<br/>
+SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+The basal principle, which was the pivot of all our previous considerations,
+was the <i>special</i> principle of relativity, <i>i.e.</i> the principle of the physical
+relativity of all <i>uniform</i> motion. Let as once more analyse its meaning
+carefully.
+</p>
+
+<p>
+It was at all times clear that, from the point of view of the idea it conveys
+to us, every motion must be considered only as a relative motion. Returning to
+the illustration we have frequently used of the embankment and the railway
+carriage, we can express the fact of the motion here taking place in the
+following two forms, both of which are equally justifiable:
+</p>
+
+<p class="letter">
+(<i>a</i>) The carriage is in motion relative to the embankment,
+</p>
+
+<p class="letter">
+(<i>b</i>) The embankment is in motion relative to the carriage.
+</p>
+
+<p>
+In (<i>a</i>) the embankment, in (<i>b</i>) the carriage, serves as the body of reference in
+our statement of the motion taking place. If it is simply a question of
+detecting or of describing the motion involved, it is in principle immaterial
+to what reference-body we refer the motion. As already mentioned, this is
+self-evident, but it must not be confused with the much more comprehensive
+statement called “the principle of relativity,” which we have taken
+as the basis of our investigations.
+</p>
+
+<p>
+The principle we have made use of not only maintains that we may equally well
+choose the carriage or the embankment as our reference-body for the description
+of any event (for this, too, is self-evident). Our principle rather asserts
+what follows: If we formulate the general laws of nature as they are obtained
+from experience, by making use of
+</p>
+
+<p class="letter">
+(<i>a</i>) the embankment as reference-body,
+</p>
+
+<p class="letter">
+(<i>b</i>) the railway carriage as reference-body,
+</p>
+
+<p>
+then these general laws of nature (<i>e.g.</i> the laws of mechanics or the law of the
+propagation of light <i>in vacuo</i>) have exactly the same form in both cases. This
+can also be expressed as follows: For the physical description of natural
+processes, neither of the reference bodies <i>K, K&#x2032;</i> is unique (lit.
+“specially marked out”) as compared with the other. Unlike the first,
+this latter statement need not of necessity hold <i>a priori;</i> it is not contained
+in the conceptions of “motion” and “reference-body” and
+derivable from them; only <i>experience</i> can decide as to its correctness or
+incorrectness.
+</p>
+
+<p>
+Up to the present, however, we have by no means maintained the equivalence of
+<i>all</i> bodies of reference <i>K</i> in connection with the formulation of natural laws.
+Our course was more on the following Iines. In the first place, we started out
+from the assumption that there exists a reference-body <i>K</i>, whose condition of
+motion is such that the Galileian law holds with respect to it: A particle
+left to itself and sufficiently far removed from all other particles moves
+uniformly in a straight line. With reference to K (Galileian reference-body)
+the laws of nature were to be as simple as possible. But in addition to K, all
+bodies of reference <i>K&#x2032;</i> should be given preference in this sense, and they
+should be exactly equivalent to <i>K</i> for the formulation of natural laws, provided
+that they are in a state of <i>uniform rectilinear and non-rotary motion</i> with
+respect to <i>K</i>; all these bodies of reference are to be regarded as Galileian
+reference-bodies. The validity of the principle of relativity was assumed only
+for these reference-bodies, but not for others (<i>e.g.</i> those possessing motion of
+a different kind). In this sense we speak of the <i>special</i> principle of
+relativity, or special theory of relativity.
+</p>
+
+<p>
+In contrast to this we wish to understand by the “general principle of
+relativity” the following statement: All bodies of reference <i>K, K&#x2032;</i>, etc.,
+are equivalent for the description of natural phenomena (formulation of the
+general laws of nature), whatever may be their state of motion. But before
+proceeding farther, it ought to be pointed out that this formulation must be
+replaced later by a more abstract one, for reasons which will become evident at
+a later stage.
+</p>
+
+<p>
+Since the introduction of the special principle of relativity has been
+justified, every intellect which strives after generalisation must feel the
+temptation to venture the step towards the general principle of relativity. But
+a simple and apparently quite reliable consideration seems to suggest that, for
+the present at any rate, there is little hope of success in such an attempt;
+Let us imagine ourselves transferred to our old friend the railway carriage,
+which is travelling at a uniform rate. As long as it is moving uniformly, the
+occupant of the carriage is not sensible of its motion, and it is for this
+reason that he can without reluctance interpret the facts of the case as
+indicating that the carriage is at rest, but the embankment in motion.
+Moreover, according to the special principle of relativity, this interpretation
+is quite justified also from a physical point of view. If the motion of the
+carriage is now changed into a non-uniform motion, as for instance by a
+powerful application of the brakes, then the occupant of the carriage
+experiences a correspondingly powerful jerk forwards. The retarded motion is
+manifested in the mechanical behaviour of bodies relative to the person in the
+railway carriage. The mechanical behaviour is different from that of the case
+previously considered, and for this reason it would appear to be impossible
+that the same mechanical laws hold relatively to the non-uniformly moving
+carriage, as hold with reference to the carriage when at rest or in uniform
+motion. At all events it is clear that the Galileian law does not hold with
+respect to the non-uniformly moving carriage. Because of this, we feel
+compelled at the present juncture to grant a kind of absolute physical reality
+to non-uniform motion, in opposition to the general principle of relativity.
+But in what follows we shall soon see that this conclusion cannot be
+maintained.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap19"></a>XIX.<br/>
+THE GRAVITATIONAL FIELD</h3>
+
+<p>
+“If we pick up a stone and then let it go, why does it fall to the
+ground?” The usual answer to this question is: “Because it is
+attracted by the earth.” Modern physics formulates the answer rather
+differently for the following reason. As a result of the more careful study of
+electromagnetic phenomena, we have come to regard action at a distance as a
+process impossible without the intervention of some intermediary medium. If,
+for instance, a magnet attracts a piece of iron, we cannot be content to regard
+this as meaning that the magnet acts directly on the iron through the
+intermediate empty space, but we are constrained to imagine—after the
+manner of Faraday—that the magnet always calls into being something
+physically real in the space around it, that something being what we call a
+“magnetic field.” In its turn this magnetic field operates on the
+piece of iron, so that the latter strives to move towards the magnet. We shall
+not discuss here the justification for this incidental conception, which is
+indeed a somewhat arbitrary one. We shall only mention that with its aid
+electromagnetic phenomena can be theoretically represented much more
+satisfactorily than without it, and this applies particularly to the
+transmission of electromagnetic waves. The effects of gravitation also are
+regarded in an analogous manner.
+</p>
+
+<p>
+The action of the earth on the stone takes place indirectly. The earth produces
+in its surrounding a gravitational field, which acts on the stone and produces
+its motion of fall. As we know from experience, the intensity of the action on
+a body dimishes according to a quite definite law, as we proceed farther and
+farther away from the earth. From our point of view this means: The law
+governing the properties of the gravitational field in space must be a
+perfectly definite one, in order correctly to represent the diminution of
+gravitational action with the distance from operative bodies. It is something
+like this: The body (<i>e.g.</i> the earth) produces a field in its immediate
+neighbourhood directly; the intensity and direction of the field at points
+farther removed from the body are thence determined by the law which governs
+the properties in space of the gravitational fields themselves.
+</p>
+
+<p>
+In contrast to electric and magnetic fields, the gravitational field exhibits a
+most remarkable property, which is of fundamental importance for what follows.
+Bodies which are moving under the sole influence of a gravitational field
+receive an acceleration, <i>which does not in the least depend either on the
+material or on the physical state of the body.</i> For instance, a piece of lead
+and a piece of wood fall in exactly the same manner in a gravitational field
+(<i>in vacuo</i>), when they start off from rest or with the same initial velocity.
+This law, which holds most accurately, can be expressed in a different form in
+the light of the following consideration.
+</p>
+
+<p>
+According to Newton’s law of motion, we have
+</p>
+
+<p>
+(Force) = (inertial mass) x (acceleration),
+</p>
+
+<p class="noindent">
+where the “inertial mass” is a characteristic constant of the
+accelerated body. If now gravitation is the cause of the acceleration, we then
+have
+</p>
+
+<p>
+(Force) = (gravitational mass) x (intensity of the gravitational field),
+</p>
+
+<p class="noindent">
+where the “gravitational mass” is likewise a characteristic constant
+for the body. From these two relations follows:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image032.jpg" style="width:100%;" alt="image032" /><br/><br/>
+</div>
+
+<p>
+If now, as we find from experience, the acceleration is to be independent of
+the nature and the condition of the body and always the same for a given
+gravitational field, then the ratio of the gravitational to the inertial mass
+must likewise be the same for all bodies. By a suitable choice of units we can
+thus make this ratio equal to unity. We then have the following law: The
+<i>gravitational</i> mass of a body is equal to its <i>inertial</i> mass.
+</p>
+
+<p>
+It is true that this important law had hitherto been recorded in mechanics, but
+it had not been <i>interpreted</i>. A satisfactory interpretation can be obtained only
+if we recognise the following fact: <i>The same</i> quality of a body manifests
+itself according to circumstances as “inertia” or as
+“weight” (lit. “heaviness”). In the following section we
+shall show to what extent this is actually the case, and how this question is
+connected with the general postulate of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap20"></a>XX.<br/>
+THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL
+POSTULATE OF RELATIVITY</h3>
+
+<p>
+We imagine a large portion of empty space, so far removed from stars and other
+appreciable masses, that we have before us approximately the conditions
+required by the fundamental law of Galilei. It is then possible to choose a
+Galileian reference-body for this part of space (world), relative to which
+points at rest remain at rest and points in motion continue permanently in
+uniform rectilinear motion. As reference-body let us imagine a spacious chest
+resembling a room with an observer inside who is equipped with apparatus.
+Gravitation naturally does not exist for this observer. He must fasten himself
+with strings to the floor, otherwise the slightest impact against the floor
+will cause him to rise slowly towards the ceiling of the room.
+</p>
+
+<p>
+To the middle of the lid of the chest is fixed externally a hook with rope
+attached, and now a “being” (what kind of a being is immaterial to
+us) begins pulling at this with a constant force. The chest together with the
+observer then begin to move “upwards” with a uniformly accelerated
+motion. In course of time their velocity will reach unheard-of values—provided
+that we are viewing all this from another reference-body which is not being
+pulled with a rope.
+</p>
+
+<p>
+But how does the man in the chest regard the Process? The acceleration of the
+chest will be transmitted to him by the reaction of the floor of the chest. He
+must therefore take up this pressure by means of his legs if he does not wish
+to be laid out full length on the floor. He is then standing in the chest in
+exactly the same way as anyone stands in a room of a home on our earth. If he
+releases a body which he previously had in his land, the accelertion of the
+chest will no longer be transmitted to this body, and for this reason the body
+will approach the floor of the chest with an accelerated relative motion. The
+observer will further convince himself <i>that the acceleration of the body
+towards the floor of the chest is always of the same magnitude, whatever kind
+of body he may happen to use for the experiment.</i>
+</p>
+
+<p>
+Relying on his knowledge of the gravitational field (as it was discussed in the
+preceding section), the man in the chest will thus come to the conclusion that
+he and the chest are in a gravitational field which is constant with regard to
+time. Of course he will be puzzled for a moment as to why the chest does not
+fall in this gravitational field. just then, however, he discovers the hook in
+the middle of the lid of the chest and the rope which is attached to it, and he
+consequently comes to the conclusion that the chest is suspended at rest in the
+gravitational field.
+</p>
+
+<p>
+Ought we to smile at the man and say that he errs in his conclusion? I do not
+believe we ought to if we wish to remain consistent; we must rather admit that
+his mode of grasping the situation violates neither reason nor known mechanical
+laws. Even though it is being accelerated with respect to the “Galileian
+space” first considered, we can nevertheless regard the chest as being at
+rest. We have thus good grounds for extending the principle of relativity to
+include bodies of reference which are accelerated with respect to each other,
+and as a result we have gained a powerful argument for a generalised postulate
+of relativity.
+</p>
+
+<p>
+We must note carefully that the possibility of this mode of interpretation
+rests on the fundamental property of the gravitational field of giving all
+bodies the same acceleration, or, what comes to the same thing, on the law of
+the equality of inertial and gravitational mass. If this natural law did not
+exist, the man in the accelerated chest would not be able to interpret the
+behaviour of the bodies around him on the supposition of a gravitational field,
+and he would not be justified on the grounds of experience in supposing his
+reference-body to be “at rest.”
+</p>
+
+<p>
+Suppose that the man in the chest fixes a rope to the inner side of the lid,
+and that he attaches a body to the free end of the rope. The result of this
+will be to stretch the rope so that it will hang “vertically”
+downwards. If we ask for an opinion of the cause of tension in the rope, the
+man in the chest will say: “The suspended body experiences a downward
+force in the gravitational field, and this is neutralised by the tension of the
+rope; what determines the magnitude of the tension of the rope is the
+<i>gravitational mass</i> of the suspended body.” On the other hand, an observer
+who is poised freely in space will interpret the condition of things thus:
+“The rope must perforce take part in the accelerated motion of the chest,
+and it transmits this motion to the body attached to it. The tension of the
+rope is just large enough to effect the acceleration of the body. That which
+determines the magnitude of the tension of the rope is the <i>inertial mass</i> of the
+body.” Guided by this example, we see that our extension of the principle
+of relativity implies the <i>necessity</i> of the law of the equality of inertial and
+gravitational mass. Thus we have obtained a physical interpretation of this
+law.
+</p>
+
+<p>
+From our consideration of the accelerated chest we see that a general theory of
+relativity must yield important results on the laws of gravitation. In point of
+fact, the systematic pursuit of the general idea of relativity has supplied the
+laws satisfied by the gravitational field. Before proceeding farther, however,
+I must warn the reader against a misconception suggested by these
+considerations. A gravitational field exists for the man in the chest, despite
+the fact that there was no such field for the co-ordinate system first chosen.
+Now we might easily suppose that the existence of a gravitational field is
+always only an <i>apparent</i> one. We might also think that, regardless of the kind
+of gravitational field which may be present, we could always choose another
+reference-body such that <i>no</i> gravitational field exists with reference to it.
+This is by no means true for all gravitational fields, but only for those of
+quite special form. It is, for instance, impossible to choose a body of
+reference such that, as judged from it, the gravitational field of the earth
+(in its entirety) vanishes.
+</p>
+
+<p>
+We can now appreciate why that argument is not convincing, which we brought
+forward against the general principle of relativity at the end of Section XVIII.
+It is certainly true that the observer in the railway carriage experiences a
+jerk forwards as a result of the application of the brake, and that he
+recognises, in this the non-uniformity of motion (retardation) of the carriage.
+But he is compelled by nobody to refer this jerk to a “real”
+acceleration (retardation) of the carriage. He might also interpret his
+experience thus: “My body of reference (the carriage) remains permanently
+at rest. With reference to it, however, there exists (during the period of
+application of the brakes) a gravitational field which is directed forwards and
+which is variable with respect to time. Under the influence of this field, the
+embankment together with the earth moves non-uniformly in such a manner that
+their original velocity in the backwards direction is continuously
+reduced.”
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap21"></a>XXI.<br/>
+IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE SPECIAL
+THEORY OF RELATIVITY UNSATISFACTORY?</h3>
+
+<p>
+We have already stated several times that classical mechanics starts out from
+the following law: Material particles sufficiently far removed from other
+material particles continue to move uniformly in a straight line or continue in
+a state of rest. We have also repeatedly emphasised that this fundamental law
+can only be valid for bodies of reference <i>K</i> which possess certain unique states
+of motion, and which are in uniform translational motion relative to each
+other. Relative to other reference-bodies <i>K</i> the law is not valid. Both in
+classical mechanics and in the special theory of relativity we therefore
+differentiate between reference-bodies <i>K</i> relative to which the recognised
+“laws of nature” can be said to hold, and reference-bodies <i>K</i> relative
+to which these laws do not hold.
+</p>
+
+<p>
+But no person whose mode of thought is logical can rest satisfied with this
+condition of things. He asks: “How does it come that certain
+reference-bodies (or their states of motion) are given priority over other
+reference-bodies (or their states of motion)? <i>What is the reason for this
+preference?</i>” In order to show clearly what I mean by this question, I
+shall make use of a comparison.
+</p>
+
+<p>
+I am standing in front of a gas range. Standing alongside of each other on the
+range are two pans so much alike that one may be mistaken for the other. Both
+are half full of water. I notice that steam is being emitted continuously from
+the one pan, but not from the other. I am surprised at this, even if I have
+never seen either a gas range or a pan before. But if I now notice a luminous
+something of bluish colour under the first pan but not under the other, I cease
+to be astonished, even if I have never before seen a gas flame. For I can only
+say that this bluish something will cause the emission of the steam, or at
+least <i>possibly</i> it may do so. If, however, I notice the bluish something in
+neither case, and if I observe that the one continuously emits steam whilst the
+other does not, then I shall remain astonished and dissatisfied until I have
+discovered some circumstance to which I can attribute the different behaviour
+of the two pans.
+</p>
+
+<p>
+Analogously, I seek in vain for a real something in classical mechanics (or in
+the special theory of relativity) to which I can attribute the different
+behaviour of bodies considered with respect to the reference systems <i>K</i> and <i>K&#x2032;</i>.<a href="#linknote-15" name="linknoteref-15" id="linknoteref-15">[15]</a> Newton saw this objection and attempted to invalidate it, but
+without success. But E. Mach recognised it most clearly of all, and because of
+this objection he claimed that mechanics must be placed on a new basis. It can
+only be got rid of by means of a physics which is conformable to the general
+principle of relativity, since the equations of such a theory hold for every
+body of reference, whatever may be its state of motion.
+</p>
+
+<p>
+<a name="linknote-15" id="linknote-15">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-15"> [15]</a><br/> The objection is of importance more
+especially when the state of motion of the reference-body is of such a nature
+that it does not require any external agency for its maintenance, <i>e.g.</i> in the
+case when the reference-body is rotating uniformly.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap22"></a>XXII.<br/>
+A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+The considerations of Section XX show that the general principle of relativity
+puts us in a position to derive properties of the gravitational field in a
+purely theoretical manner. Let us suppose, for instance, that we know the
+space-time “course” for any natural process whatsoever, as regards
+the manner in which it takes place in the Galileian domain relative to a
+Galileian body of reference <i>K</i>. By means of purely theoretical operations
+(<i>i.e.</i> simply by calculation) we are then able to find how this known
+natural process appears, as seen from a reference-body <i>K&#x2032;</i> which is
+accelerated relatively to <i>K</i>. But since a gravitational field exists with
+respect to this new body of reference <i>K&#x2032;</i>, our consideration also teaches
+us how the gravitational field influences the process studied.
+</p>
+
+<p>
+For example, we learn that a body which is in a state of uniform rectilinear
+motion with respect to <i>K</i> (in accordance with the law of Galilei) is executing
+an accelerated and in general curvilinear motion with respect to the
+accelerated reference-body <i>K&#x2032;</i> (chest). This acceleration or curvature
+corresponds to the influence on the moving body of the gravitational field
+prevailing relatively to <i>K</i>. It is known that a gravitational field influences
+the movement of bodies in this way, so that our consideration supplies us with
+nothing essentially new.
+</p>
+
+<p>
+However, we obtain a new result of fundamental importance when we carry out the
+analogous consideration for a ray of light. With respect to the Galileian
+reference-body <i>K</i>, such a ray of light is transmitted rectilinearly with the
+velocity <i>c</i>. It can easily be shown that the path of the same ray of light is no
+longer a straight line when we consider it with reference to the accelerated
+chest (reference-body <i>K&#x2032;</i>). From this we conclude, <i>that, in general, rays of
+light are propagated curvilinearly in gravitational fields.</i> In two respects
+this result is of great importance.
+</p>
+
+<p>
+In the first place, it can be compared with the reality. Although a detailed
+examination of the question shows that the curvature of light rays required by
+the general theory of relativity is only exceedingly small for the
+gravitational fields at our disposal in practice, its estimated magnitude for
+light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of
+arc. This ought to manifest itself in the following way. As seen from the
+earth, certain fixed stars appear to be in the neighbourhood of the sun, and
+are thus capable of observation during a total eclipse of the sun. At such
+times, these stars ought to appear to be displaced outwards from the sun by an
+amount indicated above, as compared with their apparent position in the sky
+when the sun is situated at another part of the heavens. The examination of the
+correctness or otherwise of this deduction is a problem of the greatest
+importance, the early solution of which is to be expected of astronomers.<a href="#linknote-16" name="linknoteref-16" id="linknoteref-16">[16]</a>
+</p>
+
+<p>
+<a name="linknote-16" id="linknote-16">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-16"> [16]</a><br/> By means of the star photographs of
+two expeditions equipped by a Joint Committee of the Royal and Royal
+Astronomical Societies, the existence of the deflection of light demanded by
+theory was first confirmed during the solar eclipse of 29th May, 1919. (Cf.
+Appendix III.)
+</p>
+
+<p>
+In the second place our result shows that, according to the general theory of
+relativity, the law of the constancy of the velocity of light in vacuo, which
+constitutes one of the two fundamental assumptions in the special theory of
+relativity and to which we have already frequently referred, cannot claim any
+unlimited validity. A curvature of rays of light can only take place when the
+velocity of propagation of light varies with position. Now we might think that
+as a consequence of this, the special theory of relativity and with it the
+whole theory of relativity would be laid in the dust. But in reality this is
+not the case. We can only conclude that the special theory of relativity cannot
+claim an unlimited domain of validity; its results hold only so long as we are
+able to disregard the influences of gravitational fields on the phenomena (<i>e.g.</i>
+of light).
+</p>
+
+<p>
+Since it has often been contended by opponents of the theory of relativity that
+the special theory of relativity is overthrown by the general theory of
+relativity, it is perhaps advisable to make the facts of the case clearer by
+means of an appropriate comparison. Before the development of electrodynamics
+the laws of electrostatics were looked upon as the laws of electricity. At the
+present time we know that electric fields can be derived correctly from
+electrostatic considerations only for the case, which is never strictly
+realised, in which the electrical masses are quite at rest relatively to each
+other, and to the co-ordinate system. Should we be justified in saying that for
+this reason electrostatics is overthrown by the field-equations of Maxwell in
+electrodynamics? Not in the least. Electrostatics is contained in
+electrodynamics as a limiting case; the laws of the latter lead directly to
+those of the former for the case in which the fields are invariable with regard
+to time. No fairer destiny could be allotted to any physical theory, than that
+it should of itself point out the way to the introduction of a more
+comprehensive theory, in which it lives on as a limiting case.
+</p>
+
+<p>
+In the example of the transmission of light just dealt with, we have seen that
+the general theory of relativity enables us to derive theoretically the
+influence of a gravitational field on the course of natural processes, the laws
+of which are already known when a gravitational field is absent. But the most
+attractive problem, to the solution of which the general theory of relativity
+supplies the key, concerns the investigation of the laws satisfied by the
+gravitational field itself. Let us consider this for a moment.
+</p>
+
+<p>
+We are acquainted with space-time domains which behave (approximately) in a
+“Galileian” fashion under suitable choice of reference-body, <i>i.e.</i>
+domains in which gravitational fields are absent. If we now refer such a domain
+to a reference-body <i>K&#x2032;</i> possessing any kind of motion, then relative to <i>K&#x2032;</i> there
+exists a gravitational field which is variable with respect to space and time.<a href="#linknote-17" name="linknoteref-17" id="linknoteref-17">[17]</a> The character of this field will of course depend on the motion
+chosen for <i>K&#x2032;.</i> According to the general theory of relativity, the general law
+of the gravitational field must be satisfied for all gravitational fields
+obtainable in this way. Even though by no means all gravitationial fields can
+be produced in this way, yet we may entertain the hope that the general law of
+gravitation will be derivable from such gravitational fields of a special kind.
+This hope has been realised in the most beautiful manner. But between the clear
+vision of this goal and its actual realisation it was necessary to surmount a
+serious difficulty, and as this lies deep at the root of things, I dare not
+withhold it from the reader. We require to extend our ideas of the space-time
+continuum still farther.
+</p>
+
+<p>
+<a name="linknote-17" id="linknote-17">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-17"> [17]</a><br/> This follows from a generalisation of
+the discussion in Section XX.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap23"></a>XXIII.<br/>
+BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE</h3>
+
+<p>
+Hitherto I have purposely refrained from speaking about the physical
+interpretation of space- and time-data in the case of the general theory of
+relativity. As a consequence, I am guilty of a certain slovenliness of
+treatment, which, as we know from the special theory of relativity, is far from
+being unimportant and pardonable. It is now high time that we remedy this
+defect; but I would mention at the outset, that this matter lays no small
+claims on the patience and on the power of abstraction of the reader.
+</p>
+
+<p>
+We start off again from quite special cases, which we have frequently used
+before. Let us consider a space time domain in which no gravitational field
+exists relative to a reference-body <i>K</i> whose state of motion has been
+suitably chosen. <i>K</i> is then a Galileian reference-body as regards the
+domain considered, and the results of the special theory of relativity hold
+relative to <i>K</i>. Let us suppose the same domain referred to a second body
+of reference <i>K&#x2032;</i>, which is rotating uniformly with respect to <i>K</i>.
+In order to fix our ideas, we shall imagine <i>K&#x2032;</i> to be in the form of a
+plane circular disc, which rotates uniformly in its own plane about its centre.
+An observer who is sitting eccentrically on the disc <i>K&#x2032;</i> is sensible of a
+force which acts outwards in a radial direction, and which would be interpreted
+as an effect of inertia (centrifugal force) by an observer who was at rest with
+respect to the original reference-body <i>K</i>. But the observer on the disc
+may regard his disc as a reference-body which is “at rest”; on the
+basis of the general principle of relativity he is justified in doing this. The
+force acting on himself, and in fact on all other bodies which are at rest
+relative to the disc, he regards as the effect of a gravitational field.
+Nevertheless, the space-distribution of this gravitational field is of a kind
+that would not be possible on Newton’s theory of gravitation.<a
+href="#linknote-18" name="linknoteref-18" id="linknoteref-18">[18]</a> But
+since the observer believes in the general theory of relativity, this does not
+disturb him; he is quite in the right when he believes that a general law of
+gravitation can be formulated—a law which not only explains the motion of
+the stars correctly, but also the field of force experienced by himself.
+</p>
+
+<p>
+<a name="linknote-18" id="linknote-18">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-18"> [18]</a><br/> The field disappears at the centre of
+the disc and increases proportionally to the distance from the centre as we
+proceed outwards.
+</p>
+
+<p>
+The observer performs experiments on his circular disc with clocks and
+measuring-rods. In doing so, it is his intention to arrive at exact definitions
+for the signification of time- and space-data with reference to the circular
+disc <i>K&#x2032;</i>, these definitions being based on his observations. What will be his
+experience in this enterprise?
+</p>
+
+<p>
+To start with, he places one of two identically constructed clocks at the
+centre of the circular disc, and the other on the edge of the disc, so that
+they are at rest relative to it. We now ask ourselves whether both clocks go at
+the same rate from the standpoint of the non-rotating Galileian reference-body
+<i>K</i>. As judged from this body, the clock at the centre of the disc has no
+velocity, whereas the clock at the edge of the disc is in motion relative to <i>K</i>
+in consequence of the rotation. According to a result obtained in Section XII,
+it follows that the latter clock goes at a rate permanently slower than that of
+the clock at the centre of the circular disc, <i>i.e.</i> as observed from <i>K</i>. It is
+obvious that the same effect would be noted by an observer whom we will imagine
+sitting alongside his clock at the centre of the circular disc. Thus on our
+circular disc, or, to make the case more general, in every gravitational field,
+a clock will go more quickly or less quickly, according to the position in
+which the clock is situated (at rest). For this reason it is not possible to
+obtain a reasonable definition of time with the aid of clocks which are
+arranged at rest with respect to the body of reference. A similar difficulty
+presents itself when we attempt to apply our earlier definition of simultaneity
+in such a case, but I do not wish to go any farther into this question.
+</p>
+
+<p>
+Moreover, at this stage the definition of the space co-ordinates also presents
+insurmountable difficulties. If the observer applies his standard measuring-rod
+(a rod which is short as compared with the radius of the disc) tangentially to
+the edge of the disc, then, as judged from the Galileian system, the length of
+this rod will be less than 1, since, according to Section XII, moving bodies
+suffer a shortening in the direction of the motion. On the other hand, the
+measuring-rod will not experience a shortening in length, as judged from <i>K</i>, if
+it is applied to the disc in the direction of the radius. If, then, the
+observer first measures the circumference of the disc with his measuring-rod
+and then the diameter of the disc, on dividing the one by the other, he will
+not obtain as quotient the familiar number &#960; = 3.14 . . ., but a larger
+number,<a href="#linknote-19" name="linknoteref-19" id="linknoteref-19">[19]</a> whereas of course, for a disc which is at rest with
+respect to <i>K</i>, this operation would yield &#960; exactly. This proves that the
+propositions of Euclidean geometry cannot hold exactly on the rotating disc,
+nor in general in a gravitational field, at least if we attribute the length 1
+to the rod in all positions and in every orientation. Hence the idea of a
+straight line also loses its meaning. We are therefore not in a position to
+define exactly the co-ordinates <i>x, y, z</i> relative to the disc by means of the
+method used in discussing the special theory, and as long as the co-ordinates
+and times of events have not been defined, we cannot assign an exact meaning to
+the natural laws in which these occur.
+</p>
+
+<p>
+<a name="linknote-19" id="linknote-19">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-19"> [19]</a><br/> Throughout this consideration we have
+to use the Galileian (non-rotating) system <i>K</i> as reference-body, since we
+may only assume the validity of the results of the special theory of relativity
+relative to <i>K</i> (relative to <i>K&#x2032;</i> a gravitational field prevails).
+</p>
+
+<p>
+Thus all our previous conclusions based on general relativity would appear to
+be called in question. In reality we must make a subtle detour in order to be
+able to apply the postulate of general relativity exactly. I shall prepare the
+reader for this in the following paragraphs.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap24"></a>XXIV.<br/>
+EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM</h3>
+
+<p>
+The surface of a marble table is spread out in front of me. I can get from any
+one point on this table to any other point by passing continuously from one
+point to a “neighbouring” one, and repeating this process a (large)
+number of times, or, in other words, by going from point to point without
+executing “jumps.” I am sure the reader will appreciate with
+sufficient clearness what I mean here by “neighbouring” and by
+“jumps” (if he is not too pedantic). We express this property of the
+surface by describing the latter as a continuum.
+</p>
+
+<p>
+Let us now imagine that a large number of little rods of equal length have been
+made, their lengths being small compared with the dimensions of the marble
+slab. When I say they are of equal length, I mean that one can be laid on any
+other without the ends overlapping. We next lay four of these little rods on
+the marble slab so that they constitute a quadrilateral figure (a square), the
+diagonals of which are equally long. To ensure the equality of the diagonals,
+we make use of a little testing-rod. To this square we add similar ones, each
+of which has one rod in common with the first. We proceed in like manner with
+each of these squares until finally the whole marble slab is laid out with
+squares. The arrangement is such, that each side of a square belongs to two
+squares and each corner to four squares.
+</p>
+
+<p>
+It is a veritable wonder that we can carry out this business without getting
+into the greatest difficulties. We only need to think of the following. If at
+any moment three squares meet at a corner, then two sides of the fourth square
+are already laid, and, as a consequence, the arrangement of the remaining two
+sides of the square is already completely determined. But I am now no longer
+able to adjust the quadrilateral so that its diagonals may be equal. If they
+are equal of their own accord, then this is an especial favour of the marble
+slab and of the little rods, about which I can only be thankfully surprised. We
+must experience many such surprises if the construction is to be successful.
+</p>
+
+<p>
+If everything has really gone smoothly, then I say that the points of the
+marble slab constitute a Euclidean continuum with respect to the little rod,
+which has been used as a “distance” (line-interval). By choosing one
+corner of a square as “origin” I can characterise every other corner
+of a square with reference to this origin by means of two numbers. I only need
+state how many rods I must pass over when, starting from the origin, I proceed
+towards the “right” and then “upwards,” in order to arrive
+at the corner of the square under consideration. These two numbers are then the
+“Cartesian co-ordinates” of this corner with reference to the
+“Cartesian co-ordinate system” which is determined by the arrangement
+of little rods.
+</p>
+
+<p>
+By making use of the following modification of this abstract experiment, we
+recognise that there must also be cases in which the experiment would be
+unsuccessful. We shall suppose that the rods “expand” by in amount
+proportional to the increase of temperature. We heat the central part of the
+marble slab, but not the periphery, in which case two of our little rods can
+still be brought into coincidence at every position on the table. But our
+construction of squares must necessarily come into disorder during the heating,
+because the little rods on the central region of the table expand, whereas
+those on the outer part do not.
+</p>
+
+<p>
+With reference to our little rods—defined as unit lengths—the marble slab is
+no longer a Euclidean continuum, and we are also no longer in the position of
+defining Cartesian co-ordinates directly with their aid, since the above
+construction can no longer be carried out. But since there are other things
+which are not influenced in a similar manner to the little rods (or perhaps not
+at all) by the temperature of the table, it is possible quite naturally to
+maintain the point of view that the marble slab is a “Euclidean
+continuum.” This can be done in a satisfactory manner by making a more
+subtle stipulation about the measurement or the comparison of lengths.
+</p>
+
+<p>
+But if rods of every kind (<i>i.e.</i> of every material) were to behave <i>in
+the same way</i> as regards the influence of temperature when they are on the
+variably heated marble slab, and if we had no other means of detecting the
+effect of temperature than the geometrical behaviour of our rods in experiments
+analogous to the one described above, then our best plan would be to assign the
+distance one to two points on the slab, provided that the ends of one of our
+rods could be made to coincide with these two points; for how else should we
+define the distance without our proceeding being in the highest measure grossly
+arbitrary? The method of Cartesian coordinates must then be discarded, and
+replaced by another which does not assume the validity of Euclidean geometry
+for rigid bodies.<a href="#linknote-20" name="linknoteref-20"
+id="linknoteref-20">[20]</a> The reader will notice that the situation depicted
+here corresponds to the one brought about by the general postulate of
+relativity (Section XXIII).
+</p>
+
+<p>
+<a name="linknote-20" id="linknote-20">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-20"> [20]</a><br/> Mathematicians have been confronted
+with our problem in the following form. If we are given a surface (<i>e.g.</i> an
+ellipsoid) in Euclidean three-dimensional space, then there exists for this
+surface a two-dimensional geometry, just as much as for a plane surface. Gauss
+undertook the task of treating this two-dimensional geometry from first
+principles, without making use of the fact that the surface belongs to a
+Euclidean continuum of three dimensions. If we imagine constructions to be made
+with rigid rods <i>in the surface</i> (similar to that above with the marble
+slab), we should find that different laws hold for these from those resulting
+on the basis of Euclidean plane geometry. The surface is not a Euclidean
+continuum with respect to the rods, and we cannot define Cartesian co-ordinates
+<i>in the surface</i>. Gauss indicated the principles according to which we can
+treat the geometrical relationships in the surface, and thus pointed out the
+way to the method of Riemann of treating multi-dimensional, non-Euclidean
+<i>continuum</i>. Thus it is that mathematicians long ago solved the formal
+problems to which we are led by the general postulate of relativity.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap25"></a>XXV.<br/>
+GAUSSIAN CO-ORDINATES</h3>
+
+<div class="fig" style="width:50%;">
+<img src="images/image033.jpg" style="width:100%;" alt="image033" /><br/><br/>
+</div>
+
+<p>
+According to Gauss, this combined analytical and geometrical mode of handling
+the problem can be arrived at in the following way. We imagine a system of
+arbitrary curves (see Fig. 4) drawn on the surface of the table. These we
+designate as <i>u</i>-curves, and we indicate each of them by means of a number. The
+Curves <i>u</i> = 1, <i>u</i> = 2 and <i>u</i> = 3 are drawn in the diagram. Between the curves <i>u</i> =
+1 and <i>u</i> = 2 we must imagine an infinitely large number to be drawn, all of
+which correspond to real numbers lying between 1 and 2. We have then a
+system of <i>u</i>-curves, and this “infinitely dense” system covers the
+whole surface of the table. These <i>u</i>-curves must not intersect each other, and
+through each point of the surface one and only one curve must pass. Thus a
+perfectly definite value of <i>u</i> belongs to every point on the surface of the
+marble slab. In like manner we imagine a system of <i>v</i>-curves drawn on the
+surface. These satisfy the same conditions as the <i>u</i>-curves, they are provided
+with numbers in a corresponding manner, and they may likewise be of arbitrary
+shape. It follows that a value of <i>u</i> and a value of <i>v</i> belong to every point on
+the surface of the table. We call these two numbers the co-ordinates of the
+surface of the table (Gaussian co-ordinates). For example, the point <i>P</i> in the
+diagram has the Gaussian co-ordinates <i>u</i> = 3, <i>v</i> = 1. Two neighbouring points <i>P</i>
+and <i>P&#x2032;</i> on the surface then correspond to the co-ordinates
+</p>
+
+<p>
+<i>P</i>: <i>u, v</i>
+</p>
+
+<p>
+<i>P&#x2032;</i>: <i>u</i> + <i>du, v</i> + <i>dv</i>,
+</p>
+
+<p class="noindent">
+where <i>du</i> and <i>dv</i> signify very small numbers. In a similar manner we may indicate
+the distance (line-interval) between <i>P</i> and <i>P&#x2032;</i>, as measured with a
+little rod, by means of the very small number <i>ds</i>. Then according to Gauss we
+have
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>g</i><sub>11</sub><i>du</i><sup>2</sup> + 2<i>g</i><sub>12</sub><i>du dv</i> +
+<i>g</i><sub>22</sub><i>dv</i><sup>2</sup>,
+</p>
+
+<p class="noindent">
+where <i>g</i><sub>11</sub>, <i>g</i><sub>12</sub>, <i>g</i><sub>22</sub>, are magnitudes which
+depend in a perfectly definite way on <i>u</i> and <i>v</i>. The magnitudes <i>g</i><sub>11</sub>,
+<i>g</i><sub>12</sub> and <i>g</i><sub>22</sub>, determine the behaviour of the rods relative
+to the <i>u</i>-curves and <i>v</i>-curves, and thus also relative to the surface of the
+table. For the case in which the points of the surface considered form a
+Euclidean continuum with reference to the measuring-rods, but only in this
+case, it is possible to draw the <i>u</i>-curves and <i>v</i>-curves and to attach numbers to
+them, in such a manner, that we simply have:
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>du</i><sup>2</sup> + <i>dv</i><sup>2</sup>
+</p>
+
+<p class="noindent">
+Under these conditions, the <i>u</i>-curves and <i>v</i>-curves are straight lines in the
+sense of Euclidean geometry, and they are perpendicular to each other. Here the
+Gaussian coordinates are simply Cartesian ones. It is clear that Gauss
+co-ordinates are nothing more than an association of two sets of numbers with
+the points of the surface considered, of such a nature that numerical values
+differing very slightly from each other are associated with neighbouring points
+“in space.”
+</p>
+
+<p>
+So far, these considerations hold for a continuum of two dimensions. But the
+Gaussian method can be applied also to a continuum of three, four or more
+dimensions. If, for instance, a continuum of four dimensions be supposed
+available, we may represent it in the following way. With every point of the
+continuum, we associate arbitrarily four numbers, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,
+<i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, which are known as “co-ordinates.”
+Adjacent points correspond to adjacent values of the coordinates. If a distance
+<i>ds</i> is associated with the adjacent points <i>P</i> and <i>P&#x2032;</i>, this distance
+being measurable and well defined from a physical point of view, then the
+following formula holds:
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>g</i><sub>11</sub><i>dx</i><sub>1</sub><sup>2</sup>
++ 2<i>g</i><sub>12</sub><i>dx</i><sub>1</sub><i>dx</i><sub>2</sub> . . . . +
+<i>g</i><sub>44</sub><i>dx</i><sub>4</sub><sup>2</sup>,
+</p>
+
+<p class="noindent">
+where the magnitudes <i>g</i><sub>11</sub>, etc., have values which vary with the
+position in the continuum. Only when the continuum is a Euclidean one is it
+possible to associate the co-ordinates <i>x</i><sub>1</sub> . . <i>x</i><sub>4</sub>. with
+the points of the continuum so that we have simply
+</p>
+
+<p>
+<i>ds</i><sup>2</sup> = <i>dx</i><sub>1</sub><sup>2</sup> +
+<i>dx</i><sub>2</sub><sup>2</sup> + <i>dx</i><sub>3</sub><sup>2</sup> +
+<i>dx</i><sub>4</sub><sup>2</sup>.
+</p>
+
+<p class="noindent">
+In this case relations hold in the four-dimensional continuum which are
+analogous to those holding in our three-dimensional measurements.
+</p>
+
+<p>
+However, the Gauss treatment for <i>ds</i><sup>2</sup> which we have given above is
+not always possible. It is only possible when sufficiently small regions of the
+continuum under consideration may be regarded as Euclidean continua. For
+example, this obviously holds in the case of the marble slab of the table and
+local variation of temperature. The temperature is practically constant for a
+small part of the slab, and thus the geometrical behaviour of the rods is
+<i>almost</i> as it ought to be according to the rules of Euclidean geometry. Hence
+the imperfections of the construction of squares in the previous section do not
+show themselves clearly until this construction is extended over a considerable
+portion of the surface of the table.
+</p>
+
+<p>
+We can sum this up as follows: Gauss invented a method for the mathematical
+treatment of continua in general, in which “size-relations”
+(“distances” between neighbouring points) are defined. To every point
+of a continuum are assigned as many numbers (Gaussian coordinates) as the
+continuum has dimensions. This is done in such a way, that only one meaning can
+be attached to the assignment, and that numbers (Gaussian coordinates) which
+differ by an indefinitely small amount are assigned to adjacent points. The
+Gaussian coordinate system is a logical generalisation of the Cartesian
+co-ordinate system. It is also applicable to non-Euclidean continua, but only
+when, with respect to the defined “size” or “distance,”
+small parts of the continuum under consideration behave more nearly like a
+Euclidean system, the smaller the part of the continuum under our notice.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap26"></a>XXVI.<br/>
+THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED AS A
+EUCLIDEAN CONTINUUM</h3>
+
+<p>
+We are now in a position to formulate more exactly the idea of Minkowski, which
+was only vaguely indicated in Section XVII. In accordance with the special theory
+of relativity, certain co-ordinate systems are given preference for the
+description of the four-dimensional, space-time continuum. We called these
+“Galileian co-ordinate systems.” For these systems, the four
+co-ordinates <i>x, y, z, t</i>, which determine an event or—in other words—a point
+of the four-dimensional continuum, are defined physically in a simple manner, as
+set forth in detail in the first part of this book. For the transition from one
+Galileian system to another, which is moving uniformly with reference to the
+first, the equations of the Lorentz transformation are valid. These last form
+the basis for the derivation of deductions from the special theory of
+relativity, and in themselves they are nothing more than the expression of the
+universal validity of the law of transmission of light for all Galileian
+systems of reference.
+</p>
+
+<p>
+Minkowski found that the Lorentz transformations satisfy the following simple
+conditions. Let us consider two neighbouring events, the relative position of
+which in the four-dimensional continuum is given with respect to a Galileian
+reference-body <i>K</i> by the space co-ordinate differences <i>dx, dy, dz</i>
+and the time-difference <i>dt</i>. With reference to a second Galileian system
+we shall suppose that the corresponding differences for these two events are
+<i>dx&#x2032;, dy&#x2032;, dz&#x2032;, dt&#x2032;</i>. Then these magnitudes always fulfill the condition.<a href="#linknote-21" name="linknoteref-21" id="linknoteref-21">[21]</a>
+</p>
+
+<p>
+<a name="linknote-21" id="linknote-21">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-21"> [21]</a><br/> Cf. Appendixes I and II. The relations
+which are derived there for the co-ordinates themselves are valid also for
+co-ordinate <i>differences</i>, and thus also for co-ordinate differentials
+(indefinitely small differences).
+</p>
+
+<p class="center">
+<i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>dt</i><sup>2</sup> = <i>dx&#x2032;</i><sup>2</sup> +
+<i>dy&#x2032;</i><sup>2</sup> + <i>dz&#x2032;</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>dt&#x2032;</i><sup>2</sup>.
+</p>
+
+<p>
+The validity of the Lorentz transformation follows from this condition. We can
+express this as follows: The magnitude
+</p>
+
+<p class="center">
+<i>ds</i><sup>2</sup> = <i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup> –
+<i>c</i><sup>2</sup> <i>dt</i><sup>2</sup>,
+</p>
+
+<p class="noindent">
+which belongs to two adjacent points of the four-dimensional space-time
+continuum, has the same value for all selected (Galileian) reference-bodies. If
+we replace <i>x, y, z</i>,
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image034.jpg" style="width:100%;" alt="image034" /><br/><br/>
+</div>
+
+<p class="noindent">
+by <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, we also obtain
+the result that
+</p>
+
+<p class="center">
+<i>ds</i><sup>2</sup> = <i>dx</i><sub>1</sub><sup>2</sup> + <i>dx</i><sub>2</sub><sup>2</sup> +
+<i>dx</i><sub>3</sub><sup>2</sup> + <i>dx</i><sub>4</sub><sup>2</sup>.
+</p>
+
+<p class="noindent">
+is independent of the choice of the body of reference. We call the magnitude <i>ds</i>
+the “distance” apart of the two events or four-dimensional points.
+</p>
+
+<p>
+Thus, if we choose as time-variable the imaginary variable
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image035.jpg" style="width:100%;" alt="image035" /><br/><br/>
+</div>
+
+<p class="noindent">
+instead of the real quantity <i>t</i>, we can regard the space-time
+contintium—accordance with the special theory of relativity—as a
+“Euclidean” four-dimensional continuum, a result which follows from
+the considerations of the preceding section.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap27"></a>XXVII.<br/>
+THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A EUCLIDEAN
+CONTINUUM</h3>
+
+<p>
+In the first part of this book we were able to make use of space-time
+co-ordinates which allowed of a simple and direct physical interpretation, and
+which, according to Section XXVI, can be regarded as four-dimensional Cartesian
+co-ordinates. This was possible on the basis of the law of the constancy of the
+velocity of light. But according to Section XXI the general theory of relativity
+cannot retain this law. On the contrary, we arrived at the result that
+according to this latter theory the velocity of light must always depend on the
+co-ordinates when a gravitational field is present. In connection with a
+specific illustration in Section XXIII, we found that the presence of a
+gravitational field invalidates the definition of the coordinates and the time,
+which led us to our objective in the special theory of relativity.
+</p>
+
+<p>
+In view of the resuIts of these considerations we are led to the conviction
+that, according to the general principle of relativity, the space-time
+continuum cannot be regarded as a Euclidean one, but that here we have the
+general case, corresponding to the marble slab with local variations of
+temperature, and with which we made acquaintance as an example of a
+two-dimensional continuum. Just as it was there impossible to construct a
+Cartesian co-ordinate system from equal rods, so here it is impossible to build
+up a system (reference-body) from rigid bodies and clocks, which shall be of
+such a nature that measuring-rods and clocks, arranged rigidly with respect to
+one another, shall indicate position and time directly. Such was the essence of
+the difficulty with which we were confronted in Section XXIII.
+</p>
+
+<p>
+But the considerations of Sections XXV and XXVI show us the way to surmount this
+difficulty. We refer the four-dimensional space-time continuum in an arbitrary
+manner to Gauss co-ordinates. We assign to every point of the continuum (event)
+four numbers, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>
+(co-ordinates), which have not the least direct physical significance, but only
+serve the purpose of numbering the points of the continuum in a definite but
+arbitrary manner. This arrangement does not even need to be of such a kind that
+we must regard <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, as
+“space” co-ordinates and <i>x</i><sub>4</sub>, as a “time”
+co-ordinate.
+</p>
+
+<p>
+The reader may think that such a description of the world would be quite
+inadequate. What does it mean to assign to an event the particular co-ordinates
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, if in themselves
+these co-ordinates have no significance? More careful consideration shows,
+however, that this anxiety is unfounded. Let us consider, for instance, a
+material point with any kind of motion. If this point had only a momentary
+existence without duration, then it would to described in space-time by a
+single system of values <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>,
+<i>x</i><sub>4</sub>. Thus its permanent existence must be characterised by an
+infinitely large number of such systems of values, the co-ordinate values of
+which are so close together as to give continuity; corresponding to the
+material point, we thus have a (uni-dimensional) line in the four-dimensional
+continuum. In the same way, any such lines in our continuum correspond to many
+points in motion. The only statements having regard to these points which can
+claim a physical existence are in reality the statements about their
+encounters. In our mathematical treatment, such an encounter is expressed in
+the fact that the two lines which represent the motions of the points in
+question have a particular system of co-ordinate values, <i>x</i><sub>1</sub>,
+<i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, in common. After mature
+consideration the reader will doubtless admit that in reality such encounters
+constitute the only actual evidence of a time-space nature with which we meet
+in physical statements.
+</p>
+
+<p>
+When we were describing the motion of a material point relative to a body of
+reference, we stated nothing more than the encounters of this point with
+particular points of the reference-body. We can also determine the
+corresponding values of the time by the observation of encounters of the body
+with clocks, in conjunction with the observation of the encounter of the hands
+of clocks with particular points on the dials. It is just the same in the case
+of space-measurements by means of measuring-rods, as a little consideration
+will show.
+</p>
+
+<p>
+The following statements hold generally: Every physical description resolves
+itself into a number of statements, each of which refers to the space-time
+coincidence of two events <i>A</i> and <i>B</i>. In terms of Gaussian co-ordinates, every
+such statement is expressed by the agreement of their four co-ordinates
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>. Thus in reality,
+the description of the time-space continuum by means of Gauss co-ordinates
+completely replaces the description with the aid of a body of reference,
+without suffering from the defects of the latter mode of description; it is not
+tied down to the Euclidean character of the continuum which has to be
+represented.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap28"></a>XXVIII.<br/>
+EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+We are now in a position to replace the provisional formulation of the general
+principle of relativity given in Section XVIII by an exact formulation. The form
+there used, “All bodies of reference <i>K, K&#x2032;</i>, etc., are equivalent for the
+description of natural phenomena (formulation of the general laws of nature),
+whatever may be their state of motion,” cannot be maintained, because the
+use of rigid reference-bodies, in the sense of the method followed in the
+special theory of relativity, is in general not possible in space-time
+description. The Gauss co-ordinate system has to take the place of the body of
+reference. The following statement corresponds to the fundamental idea of the
+general principle of relativity: “<i>All Gaussian co-ordinate systems are
+essentially equivalent for the formulation of the general laws of nature.</i>”
+</p>
+
+<p>
+We can state this general principle of relativity in still another form, which
+renders it yet more clearly intelligible than it is when in the form of the
+natural extension of the special principle of relativity. According to the
+special theory of relativity, the equations which express the general laws of
+nature pass over into equations of the same form when, by making use of the
+Lorentz transformation, we replace the space-time variables <i>x, y, z, t</i>, of a
+(Galileian) reference-body <i>K</i> by the space-time variables <i>x&#x2032;, y&#x2032;, z&#x2032;, t&#x2032;</i>, of a
+new reference-body <i>K&#x2032;</i>. According to the general theory of relativity, on the
+other hand, by application of <i>arbitrary substitutions</i> of the Gauss variables
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, the equations must
+pass over into equations of the same form; for every transformation (not only
+the Lorentz transformation) corresponds to the transition of one Gauss
+co-ordinate system into another.
+</p>
+
+<p>
+If we desire to adhere to our “old-time” three-dimensional view of
+things, then we can characterise the development which is being undergone by
+the fundamental idea of the general theory of relativity as follows: The
+special theory of relativity has reference to Galileian domains, <i>i.e.</i> to those
+in which no gravitational field exists. In this connection a Galileian
+reference-body serves as body of reference, <i>i.e.</i> a rigid body the state of
+motion of which is so chosen that the Galileian law of the uniform rectilinear
+motion of “isolated” material points holds relatively to it.
+</p>
+
+<p>
+Certain considerations suggest that we should refer the same Galileian domains
+to <i>non-Galileian</i> reference-bodies also. A gravitational field of a special kind
+is then present with respect to these bodies (cf. Sections XX and XXIII).
+</p>
+
+<p>
+In gravitational fields there are no such things as rigid bodies with Euclidean
+properties; thus the fictitious rigid body of reference is of no avail in the
+general theory of relativity. The motion of clocks is also influenced by
+gravitational fields, and in such a way that a physical definition of time
+which is made directly with the aid of clocks has by no means the same degree
+of plausibility as in the special theory of relativity.
+</p>
+
+<p>
+For this reason non-rigid reference-bodies are used, which are as a whole not
+only moving in any way whatsoever, but which also suffer alterations in form <i>ad
+lib.</i> during their motion. Clocks, for which the law of motion is of any kind,
+however irregular, serve for the definition of time. We have to imagine each of
+these clocks fixed at a point on the non-rigid reference-body. These clocks
+satisfy only the one condition, that the “readings” which are
+observed simultaneously on adjacent clocks (in space) differ from each other by
+an indefinitely small amount. This non-rigid reference-body, which might
+appropriately be termed a “reference-mollusc”, is in the main
+equivalent to a Gaussian four-dimensional co-ordinate system chosen
+arbitrarily. That which gives the “mollusc” a certain
+comprehensibility as compared with the Gauss co-ordinate system is the (really
+unjustified) formal retention of the separate existence of the
+</p>
+
+<p>
+space co-ordinates as opposed to the time co-ordinate. Every point on the
+mollusc is treated as a space-point, and every material point which is at rest
+relatively to it as at rest, so long as the mollusc is considered as
+reference-body. The general principle of relativity requires that all these
+molluscs can be used as reference-bodies with equal right and equal success in
+the formulation of the general laws of nature; the laws themselves must be
+quite independent of the choice of mollusc.
+</p>
+
+<p>
+The great power possessed by the general principle of relativity lies in the
+comprehensive limitation which is imposed on the laws of nature in consequence
+of what we have seen above.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap29"></a>XXIX.<br/>
+THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
+PRINCIPLE OF RELATIVITY</h3>
+
+<p>
+If the reader has followed all our previous considerations, he will have no
+further difficulty in understanding the methods leading to the solution of the
+problem of gravitation.
+</p>
+
+<p>
+We start off on a consideration of a Galileian domain, <i>i.e.</i> a domain in which
+there is no gravitational field relative to the Galileian reference-body <i>K</i>. The
+behaviour of measuring-rods and clocks with reference to <i>K</i> is known from the
+special theory of relativity, likewise the behaviour of “isolated”
+material points; the latter move uniformly and in straight lines.
+</p>
+
+<p>
+Now let us refer this domain to a random Gauss coordinate system or to a
+“mollusc” as reference-body <i>K&#x2032;</i>. Then with respect to <i>K&#x2032;</i> there is a
+gravitational field <i>G</i> (of a particular kind). We learn the behaviour of
+measuring-rods and clocks and also of freely-moving material points with
+reference to <i>K&#x2032;</i> simply by mathematical transformation. We interpret this
+behaviour as the behaviour of measuring-rods, clocks and material points under
+the influence of the gravitational field <i>G</i>. Hereupon we introduce a hypothesis:
+that the influence of the gravitational field on measuring-rods, clocks and
+freely-moving material points continues to take place according to the same
+laws, even in the case where the prevailing gravitational field is <i>not</i>
+derivable from the Galileian special case, simply by means of a transformation
+of co-ordinates.
+</p>
+
+<p>
+The next step is to investigate the space-time behaviour of the gravitational
+field <i>G</i>, which was derived from the Galileian special case simply by
+transformation of the coordinates. This behaviour is formulated in a law, which
+is always valid, no matter how the reference-body (mollusc) used in the
+description may be chosen.
+</p>
+
+<p>
+This law is not yet the <i>general</i> law of the gravitational field, since the
+gravitational field under consideration is of a special kind. In order to find
+out the general law-of-field of gravitation we still require to obtain a
+generalisation of the law as found above. This can be obtained without caprice,
+however, by taking into consideration the following demands:
+</p>
+
+<p class="letter">
+(<i>a</i>) The required generalisation must likewise satisfy the general postulate of
+relativity.
+</p>
+
+<p class="letter">
+(<i>b</i>) If there is any matter in the domain under consideration, only its inertial
+mass, and thus according to Section XV only its energy is of importance for its
+effect in exciting a field.
+</p>
+
+<p class="letter">
+(<i>c</i>) Gravitational field and matter together must satisfy the law of the
+conservation of energy (and of impulse).
+</p>
+
+<p>
+Finally, the general principle of relativity permits us to determine the
+influence of the gravitational field on the course of all those processes which
+take place according to known laws when a gravitational field is absent <i>i.e.</i>
+which have already been fitted into the frame of the special theory of
+relativity. In this connection we proceed in principle according to the method
+which has already been explained for measuring-rods, clocks and freely moving
+material points.
+</p>
+
+<p>
+The theory of gravitation derived in this way from the general postulate of
+relativity excels not only in its beauty; nor in removing the defect attaching
+to classical mechanics which was brought to light in Section XXI; nor in
+interpreting the empirical law of the equality of inertial and gravitational
+mass; but it has also already explained a result of observation in astronomy,
+against which classical mechanics is powerless.
+</p>
+
+<p>
+If we confine the application of the theory to the case where the gravitational
+fields can be regarded as being weak, and in which all masses move with respect
+to the coordinate system with velocities which are small compared with the
+velocity of light, we then obtain as a first approximation the Newtonian
+theory. Thus the latter theory is obtained here without any particular
+assumption, whereas Newton had to introduce the hypothesis that the force of
+attraction between mutually attracting material points is inversely
+proportional to the square of the distance between them. If we increase the
+accuracy of the calculation, deviations from the theory of Newton make their
+appearance, practically all of which must nevertheless escape the test of
+observation owing to their smallness.
+</p>
+
+<p>
+We must draw attention here to one of these deviations. According to Newton’s
+theory, a planet moves round the sun in an ellipse, which would permanently
+maintain its position with respect to the fixed stars, if we could disregard
+the motion of the fixed stars themselves and the action of the other planets
+under consideration. Thus, if we correct the observed motion of the planets for
+these two influences, and if Newton’s theory be strictly correct, we ought to
+obtain for the orbit of the planet an ellipse, which is fixed with reference to
+the fixed stars. This deduction, which can be tested with great accuracy, has
+been confirmed for all the planets save one, with the precision that is capable
+of being obtained by the delicacy of observation attainable at the present
+time. The sole exception is Mercury, the planet which lies nearest the sun.
+Since the time of Leverrier, it has been known that the ellipse corresponding
+to the orbit of Mercury, after it has been corrected for the influences
+mentioned above, is not stationary with respect to the fixed stars, but that it
+rotates exceedingly slowly in the plane of the orbit and in the sense of the
+orbital motion. The value obtained for this rotary movement of the orbital
+ellipse was 43 seconds of arc per century, an amount ensured to be correct to
+within a few seconds of arc. This effect can be explained by means of classical
+mechanics only on the assumption of hypotheses which have little probability,
+and which were devised solely for this purponse.
+</p>
+
+<p>
+On the basis of the general theory of relativity, it is found that the ellipse
+of every planet round the sun must necessarily rotate in the manner indicated
+above; that for all the planets, with the exception of Mercury, this rotation
+is too small to be detected with the delicacy of observation possible at the
+present time; but that in the case of Mercury it must amount to 43 seconds of
+arc per century, a result which is strictly in agreement with observation.
+</p>
+
+<p>
+Apart from this one, it has hitherto been possible to make only two deductions
+from the theory which admit of being tested by observation, to wit, the
+curvature of light rays by the gravitational field of the sun,<a href="#linknote-22" name="linknoteref-22" id="linknoteref-22">[22]</a>
+and a displacement of the spectral lines of light reaching us from large stars,
+as compared with the corresponding lines for light produced in an analogous
+manner terrestrially (<i>i.e.</i> by the same kind of atom).<a href="#linknote-23" name="linknoteref-23" id="linknoteref-23">[23]</a> These two
+deductions from the theory have both been confirmed.
+</p>
+
+<p>
+<a name="linknote-22" id="linknote-22">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-22"> [22]</a><br/> First observed by Eddington and others
+in 1919. (Cf. Appendix III).
+</p>
+
+<p>
+<a name="linknote-23" id="linknote-23">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-23"> [23]</a><br/> Established by Adams in 1924. (Cf. p.
+132)
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="part03"></a>PART III: CONSIDERATIONS ON THE UNIVERSE AS A
+WHOLE</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap30"></a>XXX.<br/>
+COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY</h3>
+
+<p>
+Part from the difficulty discussed in Section XXI, there is a second fundamental
+difficulty attending classical celestial mechanics, which, to the best of my
+knowledge, was first discussed in detail by the astronomer Seeliger. If we
+ponder over the question as to how the universe, considered as a whole, is to
+be regarded, the first answer that suggests itself to us is surely this: As
+regards space (and time) the universe is infinite. There are stars everywhere,
+so that the density of matter, although very variable in detail, is
+nevertheless on the average everywhere the same. In other words: However far we
+might travel through space, we should find everywhere an attenuated swarm of
+fixed stars of approrimately the same kind and density.
+</p>
+
+<p>
+This view is not in harmony with the theory of Newton. The latter theory rather
+requires that the universe should have a kind of centre in which the density of
+the stars is a maximum, and that as we proceed outwards from this centre the
+group-density of the stars should diminish, until finally, at great distances,
+it is succeeded by an infinite region of emptiness. The stellar universe ought
+to be a finite island in the infinite ocean of space.<a href="#linknote-24" name="linknoteref-24" id="linknoteref-24">[24]</a>
+</p>
+
+<p>
+<a name="linknote-24" id="linknote-24">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-24"> [24]</a><br/> <i>Proof</i>—According to the
+theory of Newton, the number of “lines of force” which come from
+infinity and terminate in a mass m is proportional to the mass <i>m</i>. If, on
+the average, the mass density &#961;<sub>0 </sub>is constant throughout the
+universe, then a sphere of volume <i>V</i> will enclose the average mass
+&#961;<sub>0</sub><i>V</i>. Thus the number of lines of force passing through
+the surface <i>F</i> of the sphere into its interior is proportional to
+&#961;<sub>0</sub><i>V</i>. For unit area of the surface of the sphere the
+number of lines of force which enters the sphere is thus proportional to
+&#961;<sub>0</sub><i>V/F</i> or to &#961;<sub>0</sub><i>R</i>. Hence the
+intensity of the field at the surface would ultimately become infinite with
+increasing radius <i>R</i> of the sphere, which is impossible.
+</p>
+
+<p>
+This conception is in itself not very satisfactory. It is still less
+satisfactory because it leads to the result that the light emitted by the stars
+and also individual stars of the stellar system are perpetually passing out
+into infinite space, never to return, and without ever again coming into
+interaction with other objects of nature. Such a finite material universe would
+be destined to become gradually but systematically impoverished.
+</p>
+
+<p>
+In order to escape this dilemma, Seeliger suggested a modification of Newton’s
+law, in which he assumes that for great distances the force of attraction
+between two masses diminishes more rapidly than would result from the inverse
+square law. In this way it is possible for the mean density of matter to be
+constant everywhere, even to infinity, without infinitely large gravitational
+fields being produced. We thus free ourselves from the distasteful conception
+that the material universe ought to possess something of the nature of a
+centre. Of course we purchase our emancipation from the fundamental
+difficulties mentioned, at the cost of a modification and complication of
+Newton’s law which has neither empirical nor theoretical foundation. We can
+imagine innumerable laws which would serve the same purpose, without our being
+able to state a reason why one of them is to be preferred to the others; for
+any one of these laws would be founded just as little on more general
+theoretical principles as is the law of Newton.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap31"></a>XXXI.<br/>
+THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED”
+UNIVERSE</h3>
+
+<p>
+But speculations on the structure of the universe also move in quite another
+direction. The development of non-Euclidean geometry led to the recognition of
+the fact, that we can cast doubt on the <i>infiniteness</i> of our space without
+coming into conflict with the laws of thought or with experience (Riemann,
+Helmholtz). These questions have already been treated in detail and with
+unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on
+them briefly here.
+</p>
+
+<p>
+In the first place, we imagine an existence in two dimensional space. Flat
+beings with flat implements, and in particular flat rigid measuring-rods, are
+free to move in a <i>plane</i>. For them nothing exists outside of this plane: that
+which they observe to happen to themselves and to their flat “things”
+is the all-inclusive reality of their plane. In particular, the constructions
+of plane Euclidean geometry can be carried out by means of the rods <i>e.g.</i> the
+lattice construction, considered in Section XXIV. In contrast to ours, the
+universe of these beings is two-dimensional; but, like ours, it extends to
+infinity. In their universe there is room for an infinite number of identical
+squares made up of rods, <i>i.e.</i> its volume (surface) is infinite. If these beings
+say their universe is “plane,” there is sense in the statement,
+because they mean that they can perform the constructions of plane Euclidean
+geometry with their rods. In this connection the individual rods always
+represent the same distance, independently of their position.
+</p>
+
+<p>
+Let us consider now a second two-dimensional existence, but this time on a
+spherical surface instead of on a plane. The flat beings with their
+measuring-rods and other objects fit exactly on this surface and they are
+unable to leave it. Their whole universe of observation extends exclusively
+over the surface of the sphere. Are these beings able to regard the geometry of
+their universe as being plane geometry and their rods withal as the realisation
+of “distance”? They cannot do this. For if they attempt to realise a
+straight line, they will obtain a curve, which we “three-dimensional
+beings” designate as a great circle, <i>i.e.</i> a self-contained line of
+definite finite length, which can be measured up by means of a measuring-rod.
+Similarly, this universe has a finite area that can be compared with the area,
+of a square constructed with rods. The great charm resulting from this
+consideration lies in the recognition of the fact that <i>the universe of these
+beings is finite and yet has no limits.</i>
+</p>
+
+<p>
+But the spherical-surface beings do not need to go on a world-tour in order to
+perceive that they are not living in a Euclidean universe. They can convince
+themselves of this on every part of their “world,” provided they do
+not use too small a piece of it. Starting from a point, they draw
+“straight lines” (arcs of circles as judged in three dimensional
+space) of equal length in all directions. They will call the line joining the
+free ends of these lines a “circle.” For a plane surface, the ratio
+of the circumference of a circle to its diameter, both lengths being measured
+with the same rod, is, according to Euclidean geometry of the plane, equal to a
+constant value &#960;, which is independent of the diameter of the circle. On
+their spherical surface our flat beings would find for this ratio the value
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image036.jpg" style="width:100%;" alt="image036" /><br/><br/>
+</div>
+
+<p class="noindent">
+<i>i.e.</i> a smaller value than &#960;, the difference being the more considerable,
+the greater is the radius of the circle in comparison with the radius <i>R</i> of the
+“world-sphere.” By means of this relation the spherical beings can
+determine the radius of their universe (“world”), even when only a
+relatively small part of their worldsphere is available for their measurements.
+But if this part is very small indeed, they will no longer be able to
+demonstrate that they are on a spherical “world” and not on a
+Euclidean plane, for a small part of a spherical surface differs only slightly
+from a piece of a plane of the same size.
+</p>
+
+<p>
+Thus if the spherical surface beings are living on a planet of which the solar
+system occupies only a negligibly small part of the spherical universe, they
+have no means of determining whether they are living in a finite or in an
+infinite universe, because the “piece of universe” to which they
+have access is in both cases practically plane, or Euclidean. It follows
+directly from this discussion, that for our sphere-beings the circumference of
+a circle first increases with the radius until the “circumference of the
+universe” is reached, and that it thenceforward gradually decreases to
+zero for still further increasing values of the radius. During this process the
+area of the circle continues to increase more and more, until finally it
+becomes equal to the total area of the whole “world-sphere.”
+</p>
+
+<p>
+Perhaps the reader will wonder why we have placed our “beings” on a
+sphere rather than on another closed surface. But this choice has its
+justification in the fact that, of all closed surfaces, the sphere is unique in
+possessing the property that all points on it are equivalent. I admit that the
+ratio of the circumference <i>c</i> of a circle to its radius <i>r</i> depends
+on <i>r</i>, but for a given value of <i>r</i> it is the same for all points of
+the “worldsphere”; in other words, the “world-sphere”
+is a “surface of constant curvature.”
+</p>
+
+<p>
+To this two-dimensional sphere-universe there is a three-dimensional analogy,
+namely, the three-dimensional spherical space which was discovered by Riemann.
+its points are likewise all equivalent. It possesses a finite volume, which is
+determined by its “radius” (2&#960;<sup>2</sup><i>R</i><sup>3</sup>). Is it
+possible to imagine a spherical space? To imagine a space means nothing else
+than that we imagine an epitome of our “space” experience, <i>i.e.</i> of
+experience that we can have in the movement of “rigid” bodies. In
+this sense we <i>can</i> imagine a spherical space.
+</p>
+
+<p>
+Suppose we draw lines or stretch strings in all directions from a point, and
+mark off from each of these the distance <i>r</i> with a measuring-rod. All the free
+end-points of these lengths lie on a spherical surface. We can specially
+measure up the area (<i>F</i>) of this surface by means of a square made up of
+measuring-rods. If the universe is Euclidean, then
+<i>F</i> = 4&#960;<i>r</i><sup>2</sup>; if it is spherical, then <i>F</i> is always less
+than 4&#960;<i>r</i><sup>2</sup>. With increasing values of <i>r, F</i> increases from zero
+up to a maximum value which is determined by the “world-radius,” but
+for still further increasing values of <i>r</i>, the area gradually diminishes to
+zero. At first, the straight lines which radiate from the starting point
+diverge farther and farther from one another, but later they approach each
+other, and finally they run together again at a “counter-point” to
+the starting point. Under such conditions they have traversed the whole
+spherical space. It is easily seen that the three-dimensional spherical space
+is quite analogous to the two-dimensional spherical surface. It is finite (<i>i.e.</i>
+of finite volume), and has no bounds.
+</p>
+
+<p>
+It may be mentioned that there is yet another kind of curved space:
+“elliptical space.” It can be regarded as a curved space in which the
+two “counter-points” are identical (indistinguishable from each
+other). An elliptical universe can thus be considered to some extent as a
+curved universe possessing central symmetry.
+</p>
+
+<p>
+It follows from what has been said, that closed spaces without limits are
+conceivable. From amongst these, the spherical space (and the elliptical)
+excels in its simplicity, since all points on it are equivalent. As a result of
+this discussion, a most interesting question arises for astronomers and
+physicists, and that is whether the universe in which we live is infinite, or
+whether it is finite in the manner of the spherical universe. Our experience is
+far from being sufficient to enable us to answer this question. But the general
+theory of relativity permits of our answering it with a moderate degree of
+certainty, and in this connection the difficulty mentioned in Section XXX finds
+its solution.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap32"></a>XXXII.<br/>
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY</h3>
+
+<p>
+According to the general theory of relativity, the geometrical properties of
+space are not independent, but they are determined by matter. Thus we can draw
+conclusions about the geometrical structure of the universe only if we base our
+considerations on the state of the matter as being something that is known. We
+know from experience that, for a suitably chosen co-ordinate system, the
+velocities of the stars are small as compared with the velocity of transmission
+of light. We can thus as a rough approximation arrive at a conclusion as to the
+nature of the universe as a whole, if we treat the matter as being at rest.
+</p>
+
+<p>
+We already know from our previous discussion that the behaviour of
+measuring-rods and clocks is influenced by gravitational fields, <i>i.e.</i> by the
+distribution of matter. This in itself is sufficient to exclude the possibility
+of the exact validity of Euclidean geometry in our universe. But it is
+conceivable that our universe differs only slightly from a Euclidean one, and
+this notion seems all the more probable, since calculations show that the
+metrics of surrounding space is influenced only to an exceedingly small extent
+by masses even of the magnitude of our sun. We might imagine that, as regards
+geometry, our universe behaves analogously to a surface which is irregularly
+curved in its individual parts, but which nowhere departs appreciably from a
+plane: something like the rippled surface of a lake. Such a universe might
+fittingly be called a quasi-Euclidean universe. As regards its space it would
+be infinite. But calculation shows that in a quasi-Euclidean universe the
+average density of matter would necessarily be <i>nil</i>. Thus such a universe could
+not be inhabited by matter everywhere; it would present to us that
+unsatisfactory picture which we portrayed in Section XXX.
+</p>
+
+<p>
+If we are to have in the universe an average density of matter which differs
+from zero, however small may be that difference, then the universe cannot be
+quasi-Euclidean. On the contrary, the results of calculation indicate that if
+matter be distributed uniformly, the universe would necessarily be spherical
+(or elliptical). Since in reality the detailed distribution of matter is not
+uniform, the real universe will deviate in individual parts from the spherical,
+<i>i.e.</i> the universe will be quasi-spherical. But it will be necessarily finite.
+In fact, the theory supplies us with a simple connection<a href="#linknote-25" name="linknoteref-25" id="linknoteref-25">[25]</a> between
+the space-expanse of the universe and the average density of matter in it.
+</p>
+
+<p>
+<a name="linknote-25" id="linknote-25">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-25"> [25]</a><br/> For the radius <i>R</i> of the
+universe we obtain the equation
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image037.jpg" style="width:100%;" alt="image037" /><br/><br/>
+</div>
+
+<p class="footnote">
+The use of the C.G.S. system in this equation gives 2/k = 1.08 x
+10<sup>27</sup>; &#961; is the average density of the matter and <i>k</i> is a
+constant connected with the Newtonian constant of gravitation.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap33"></a>APPENDICES</h3>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap34"></a>APPENDIX I<br/>
+SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION<br/>
+(SUPPLEMENTARY TO SECTION XI)</h3>
+
+<p>
+For the relative orientation of the co-ordinate systems indicated in Fig. 2,
+the <i>x</i>-axes of both systems permanently coincide. In the present case we can
+divide the problem into parts by considering first only events which are
+localised on the <i>x</i>-axis. Any such event is represented with respect to the
+co-ordinate system <i>K</i> by the abscissa <i>x</i> and the time <i>t</i>, and with respect to the
+system <i>K&#x2032;</i> by the abscissa <i>x&#x2032;</i> and the time <i>t&#x2032;</i>. We require to find <i>x&#x2032;</i> and <i>t&#x2032;</i> when
+<i>x</i> and <i>t</i> are given.
+</p>
+
+<p>
+A light-signal, which is proceeding along the positive axis of <i>x</i>, is
+transmitted according to the equation
+</p>
+
+<p class="center">
+<i>x</i> = <i>ct</i>
+</p>
+
+<p class="noindent">
+or
+</p>
+
+<p class="center">
+<i>x</i> – <i>ct</i> = 0 . . . . . (1).
+</p>
+
+<p class="noindent">
+Since the same light-signal has to be transmitted relative to <i>K&#x2032;</i> with the
+velocity <i>c</i>, the propagation relative to the system <i>K&#x2032;</i> will be represented by
+the analogous formula
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> – <i>ct&#x2032;</i> = 0 . . . . . (2)
+</p>
+
+<p class="noindent">
+Those space-time points (events) which satisfy (1) must also satisfy (2).
+Obviously this will be the case when the relation
+</p>
+
+<p class="center">
+(<i>x&#x2032;</i> – <i>ct&#x2032;</i>) = &#955;(<i>x</i> – <i>ct</i>) . . . (3).
+</p>
+
+<p class="noindent">
+is fulfilled in general, where &#955; indicates a constant; for, according to
+(3), the disappearance of (<i>x</i> – <i>ct</i>) involves the disappearance of (<i>x&#x2032;</i> – <i>ct&#x2032;</i>).
+</p>
+
+<p>
+If we apply quite similar considerations to light rays which are being
+transmitted along the negative <i>x</i>-axis, we obtain the condition
+</p>
+
+<p class="center">
+(<i>x&#x2032;</i> + <i>ct&#x2032;</i>) = (<i>x + ct</i>) . . . (4).
+</p>
+
+<p>
+By adding (or subtracting) equations (3) and (4), and introducing for
+convenience the constants <i>a</i> and <i>b</i> in place of the constants &#955; and &#956; where
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image038.jpg" style="width:100%;" alt="image038" /><br/><br/>
+</div>
+
+<p class="noindent">
+and
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image039.jpg" style="width:100%;" alt="image039" /><br/><br/>
+</div>
+
+<p class="noindent">
+we obtain the equations
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image040.jpg" style="width:100%;" alt="image040" /><br/><br/>
+</div>
+
+<p>
+We should thus have the solution of our problem, if the constants <i>a</i> and <i>b</i> were
+known. These result from the following discussion.
+</p>
+
+<p>
+For the origin of <i>K&#x2032;</i> we have permanently <i>x&#x2032;</i> = 0, and hence according to the
+first of the equations (5)
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image041.jpg" style="width:100%;" alt="image041" /><br/><br/>
+</div>
+
+<p>
+If we call <i>v</i> the velocity with which the origin of <i>K&#x2032;</i> is moving relative to <i>K</i>,
+we then have
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image042.jpg" style="width:100%;" alt="image042" /><br/><br/>
+</div>
+
+<p>
+The same value <i>v</i> can be obtained from equations (5), if we calculate the
+velocity of another point of <i>K&#x2032;</i> relative to <i>K</i>, or the velocity (directed
+towards the negative <i>x</i>-axis) of a point of <i>K</i> with respect to <i>K&#x2032;</i>. In short, we
+can designate <i>v</i> as the relative velocity of the two systems.
+</p>
+
+<p>
+Furthermore, the principle of relativity teaches us that, as judged from K, the
+length of a unit measuring-rod which is at rest with reference to <i>K&#x2032;</i> must be
+exactly the same as the length, as judged from <i>K&#x2032;</i>, of a unit measuring-rod
+which is at rest relative to <i>K</i>. In order to see how the points of the <i>x&#x2032;</i>-axis
+appear as viewed from <i>K</i>, we only require to take a “snapshot” of <i>K&#x2032;</i>
+from <i>K</i>; this means that we have to insert a particular value of <i>t</i> (time of <i>K</i>),
+<i>e.g.</i> <i>t</i> = 0. For this value of <i>t</i> we then obtain from the first of the equations
+(5)
+</p>
+
+<p class="center">
+<i>x&#x2032;</i> = <i>ax</i>
+</p>
+
+<p>
+Two points of the <i>x&#x2032;</i>-axis which are separated by the distance &#916;<i>x&#x2032;</i> = 1 when
+measured in the <i>K&#x2032;</i> system are thus separated in our instantaneous photograph by
+the distance
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image043.jpg" style="width:100%;" alt="image043" /><br/><br/>
+</div>
+
+<p>
+But if the snapshot be taken from <i>K&#x2032;</i>(<i>t&#x2032;</i> = 0), and if we eliminate <i>t</i> from the
+equations (5), taking into account the expression (6), we obtain
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image044.jpg" style="width:100%;" alt="image044" /><br/><br/>
+</div>
+
+<p>
+From this we conclude that two points on the <i>x</i>-axis separated by the distance 1
+(relative to <i>K</i>) will be represented on our snapshot by the distance
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image045.jpg" style="width:100%;" alt="image045" /><br/><br/>
+</div>
+
+<p>
+But from what has been said, the two snapshots must be identical; hence &#916;<i>x</i>
+in (7) must be equal to &#916;<i>x&#x2032;</i> in (7<i>a</i>), so that we obtain
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image046.jpg" style="width:100%;" alt="image046" /><br/><br/>
+</div>
+
+<p>
+The equations (6) and (7<i>b</i>) determine the constants <i>a</i> and <i>b</i>. By inserting the
+values of these constants in (5), we obtain the first and the fourth of the
+equations given in Section XI.
+</p>
+
+<div class="fig" style="width:50%;">
+<img src="images/image047.jpg" style="width:100%;" alt="image047" /><br/><br/>
+</div>
+
+<p>
+Thus we have obtained the Lorentz transformation for events on the <i>x</i>-axis. It
+satisfies the condition
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup> = <i>x</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>t</i><sup>2</sup> . . . . . . (8a).
+</p>
+
+<p>
+The extension of this result, to include events which take place outside the
+<i>x</i>-axis, is obtained by retaining equations (8) and supplementing them by the
+relations
+</p>
+
+<div class="fig" style="width:60%;">
+<img src="images/image048.jpg" style="width:100%;" alt="image048" /><br/><br/>
+</div>
+
+<p class="noindent">
+In this way we satisfy the postulate of the constancy of the velocity of light
+<i>in vacuo</i> for rays of light of arbitrary direction, both for the system <i>K</i> and
+for the system <i>K&#x2032;</i>. This may be shown in the following manner.
+</p>
+
+<p>
+We suppose a light-signal sent out from the origin of <i>K</i> at the time <i>t</i> = 0. It
+will be propagated according to the equation
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image049.jpg" style="width:100%;" alt="image049" /><br/><br/>
+</div>
+
+<p class="noindent">
+or, if we square this equation, according to the equation
+</p>
+
+<p class="center">
+<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t</i><sup>2</sup> = 0
+. . . . . (10).
+</p>
+
+<p>
+It is required by the law of propagation of light, in conjunction with the
+postulate of relativity, that the transmission of the signal in question should
+take place—as judged from <i>K&#x2032;</i>—in accordance with the corresponding formula
+</p>
+
+<p class="center">
+<i>r&#x2032;</i> = <i>ct&#x2032;</i>
+</p>
+
+<p class="noindent">
+or,
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= 0 . . . . . . (10<i>a</i>).
+</p>
+
+<p class="noindent">
+In order that equation (10<i>a</i>) may be a consequence of equation (10), we must
+have
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= &#963; (<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> –
+<i>c</i><sup>2</sup><i>t</i><sup>2</sup>) (11).
+</p>
+
+<p>
+Since equation (8<i>a</i>) must hold for points on the <i>x</i>-axis, we thus have &#963; = 1. It
+is easily seen that the Lorentz transformation really satisfies equation (11)
+for &#963; = 1; for (11) is a consequence of (8<i>a</i>) and (9), and hence also of (8) and
+(9). We have thus derived the Lorentz transformation.
+</p>
+
+<p>
+The Lorentz transformation represented by (8) and (9) still requires to be
+generalised. Obviously it is immaterial whether the axes of <i>K&#x2032;</i> be chosen so
+that they are spatially parallel to those of <i>K</i>. It is also not essential that
+the velocity of translation of <i>K&#x2032;</i> with respect to <i>K</i> should be in the direction
+of the <i>x</i>-axis. A simple consideration shows that we are able to construct the
+Lorentz transformation in this general sense from two kinds of transformations,
+viz. from Lorentz transformations in the special sense and from purely spatial
+transformations. which corresponds to the replacement of the rectangular
+co-ordinate system by a new system with its axes pointing in other directions.
+</p>
+
+<p>
+Mathematically, we can characterise the generalised Lorentz transformation thus:
+</p>
+
+<p>
+It expresses <i>x&#x2032;, y&#x2032;, x&#x2032;, t&#x2032;</i>, in terms of linear homogeneous functions of <i>x, y,
+x, t</i>, of such a kind that the relation
+</p>
+
+<p class="center">
+<i>x&#x2032;</i><sup>2</sup> + <i>y&#x2032;</i><sup>2</sup> + <i>z&#x2032;</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t&#x2032;</i><sup>2</sup>
+= <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> – <i>c</i><sup>2</sup><i>t</i><sup>2</sup>
+(11<i>a</i>).
+</p>
+
+<p>
+is satisficd identically. That is to say: If we substitute their expressions in
+<i>x, y, x, t</i>, in place of <i>x&#x2032;, y&#x2032;, x&#x2032;, t&#x2032;</i>, on the left-hand side, then the
+left-hand side of (11<i>a</i>) agrees with the right-hand side.
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap35"></a>APPENDIX II<br/>
+MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)<br/>
+(SUPPLEMENTARY TO SECTION XVII)</h3>
+
+<p>
+We can characterise the Lorentz transformation still more simply if we
+introduce the imaginary
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image031.jpg" style="width:100%;" alt="image031" /><br/><br/>
+</div>
+
+<p class="noindent">
+in place of <i>t</i>, as time-variable. If, in accordance with this, we insert
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image050.jpg" style="width:100%;" alt="image050" /><br/><br/>
+</div>
+
+<p class="noindent">
+and similarly for the accented system <i>K&#x2032;</i>, then the condition which is
+identically satisfied by the transformation can be expressed thus:
+</p>
+
+<p class="center">
+<i>x</i><sub>1</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>2</sub>&#x2032;<sup>2</sup> +
+<i>x</i><sub>3</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>4</sub>&#x2032;<sup>2</sup> =
+<i>x</i><sub>1</sub><sup>2</sup> + <i>x</i><sub>2</sub><sup>2</sup> +
+<i>x</i><sub>3</sub><sup>2</sup> + <i>x</i><sub>4</sub><sup>2 </sup>(12).
+</p>
+
+<p>
+That is, by the afore-mentioned choice of “coordinates,” (11<i>a</i>) [see
+the end of Appendix II] is transformed into this equation.
+</p>
+
+<p>
+We see from (12) that the imaginary time co-ordinate <i>x</i><sub>4</sub>, enters into
+the condition of transformation in exactly the same way as the space
+co-ordinates <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>. It is due to this
+fact that, according to the theory of relativity, the “time”
+<i>x</i><sub>4</sub>, enters into natural laws in the same form as the space co
+ordinates <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>.
+</p>
+
+<p>
+A four-dimensional continuum described by the “co-ordinates”
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>, <i>x</i><sub>4</sub>, was called
+“world” by Minkowski, who also termed a point-event a
+“world-point.” From a “happening” in three-dimensional
+space, physics becomes, as it were, an “existence” in the
+four-dimensional “world.”
+</p>
+
+<p>
+This four-dimensional “world” bears a close similarity to the
+three-dimensional “space” of (Euclidean) analytical geometry. If we
+introduce into the latter a new Cartesian co-ordinate system (<i>x&#x2032;</i><sub>1</sub>,
+<i>x&#x2032;</i><sub>2</sub>, <i>x&#x2032;</i><sub>3</sub>) with the same origin, then <i>x&#x2032;</i><sub>1</sub>,
+<i>x&#x2032;</i><sub>2</sub>, <i>x&#x2032;</i><sub>3</sub>, are linear homogeneous functions of
+<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub> which identically satisfy the
+equation
+</p>
+
+<p class="center">
+<i>x</i><sub>1</sub>&#x2032;<sup>2</sup> + <i>x</i><sub>2</sub>&#x2032;<sup>2</sup> +
+<i>x</i><sub>3</sub>&#x2032;<sup>2</sup> = <i>x</i><sub>1</sub><sup>2</sup> +
+<i>x</i><sub>2</sub><sup>2</sup> + <i>x</i><sub>3</sub><sup>2</sup>
+</p>
+
+<p class="noindent">
+The analogy with (12) is a complete one. We can regard Minkowski’s
+“world” in a formal manner as a four-dimensional Euclidean space
+(with an imaginary time coordinate); the Lorentz transformation corresponds
+to a “rotation” of the co-ordinate system in the four-dimensional
+“world.”
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap36"></a>APPENDIX III<br/>
+THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY</h3>
+
+<p>
+From a systematic theoretical point of view, we may imagine the process of
+evolution of an empirical science to be a continuous process of induction.
+Theories are evolved and are expressed in short compass as statements of a
+large number of individual observations in the form of empirical laws, from
+which the general laws can be ascertained by comparison. Regarded in this way,
+the development of a science bears some resemblance to the compilation of a
+classified catalogue. It is, as it were, a purely empirical enterprise.
+</p>
+
+<p>
+But this point of view by no means embraces the whole of the actual process;
+for it slurs over the important part played by intuition and deductive thought
+in the development of an exact science. As soon as a science has emerged from
+its initial stages, theoretical advances are no longer achieved merely by a
+process of arrangement. Guided by empirical data, the investigator rather
+develops a system of thought which, in general, is built up logically from a
+small number of fundamental assumptions, the so-called axioms. We call such a
+system of thought a <i>theory</i>. The theory finds the justification for its
+existence in the fact that it correlates a large number of single observations,
+and it is just here that the “truth” of the theory lies.
+</p>
+
+<p>
+Corresponding to the same complex of empirical data, there may be several
+theories, which differ from one another to a considerable extent. But as
+regards the deductions from the theories which are capable of being tested, the
+agreement between the theories may be so complete that it becomes difficult to
+find any deductions in which the two theories differ from each other. As an
+example, a case of general interest is available in the province of biology, in
+the Darwinian theory of the development of species by selection in the struggle
+for existence, and in the theory of development which is based on the
+hypothesis of the hereditary transmission of acquired characters.
+</p>
+
+<p>
+We have another instance of far-reaching agreement between the deductions from
+two theories in Newtonian mechanics on the one hand, and the general theory of
+relativity on the other. This agreement goes so far, that up to the present we
+have been able to find only a few deductions from the general theory of
+relativity which are capable of investigation, and to which the physics of
+pre-relativity days does not also lead, and this despite the profound
+difference in the fundamental assumptions of the two theories. In what follows,
+we shall again consider these important deductions, and we shall also discuss
+the empirical evidence appertaining to them which has hitherto been obtained.
+</p>
+
+<h4>
+(<i>a</i>) Motion of the Perihelion of Mercury
+</h4>
+
+<p>
+According to Newtonian mechanics and Newton’s law of gravitation, a planet
+which is revolving round the sun would describe an ellipse round the latter,
+or, more correctly, round the common centre of gravity of the sun and the
+planet. In such a system, the sun, or the common centre of gravity, lies in one
+of the foci of the orbital ellipse in such a manner that, in the course of a
+planet-year, the distance sun-planet grows from a minimum to a maximum, and
+then decreases again to a minimum. If instead of Newton’s law we insert a
+somewhat different law of attraction into the calculation, we find that,
+according to this new law, the motion would still take place in such a manner
+that the distance sun-planet exhibits periodic variations; but in this case the
+angle described by the line joining sun and planet during such a period (from
+perihelion—closest proximity to the sun—to perihelion) would differ from 360&deg;.
+The line of the orbit would not then be a closed one but in the course of time
+it would fill up an annular part of the orbital plane, viz. between the circle
+of least and the circle of greatest distance of the planet from the sun.
+</p>
+
+<p>
+According also to the general theory of relativity, which differs of course
+from the theory of Newton, a small variation from the Newton-Kepler motion of a
+planet in its orbit should take place, and in such away, that the angle
+described by the radius sun-planet between one perhelion and the next should
+exceed that corresponding to one complete revolution by an amount given by
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image051.jpg" style="width:100%;" alt="image051" /><br/><br/>
+</div>
+
+<p>
+(<i>N.B.</i>—One complete revolution corresponds to the angle 2&#960; in the absolute
+angular measure customary in physics, and the above expression given the amount
+by which the radius sun-planet exceeds this angle during the interval between
+one perihelion and the next.) In this expression <i>a</i> represents the major
+semi-axis of the ellipse, <i>e</i> its eccentricity, <i>c</i> the velocity of light, and <i>T</i>
+the period of revolution of the planet. Our result may also be stated as
+follows: According to the general theory of relativity, the major axis of the
+ellipse rotates round the sun in the same sense as the orbital motion of the
+planet. Theory requires that this rotation should amount to 43 seconds of arc
+per century for the planet Mercury, but for the other Planets of our solar
+system its magnitude should be so small that it would necessarily escape
+detection.<a href="#linknote-26" name="linknoteref-26" id="linknoteref-26">[26]</a>
+</p>
+
+<p>
+<a name="linknote-26" id="linknote-26">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-26"> [26]</a><br/> Especially since the next planet Venus
+has an orbit that is almost an exact circle, which makes it more difficult to
+locate the perihelion with precision.
+</p>
+
+<p>
+In point of fact, astronomers have found that the theory of Newton does not
+suffice to calculate the observed motion of Mercury with an exactness
+corresponding to that of the delicacy of observation attainable at the present
+time. After taking account of all the disturbing influences exerted on Mercury
+by the remaining planets, it was found (Leverrier: 1859; and Newcomb: 1895)
+that an unexplained perihelial movement of the orbit of Mercury remained over,
+the amount of which does not differ sensibly from the above mentioned +43
+seconds of arc per century. The uncertainty of the empirical result amounts to
+a few seconds only.
+</p>
+
+<h4>
+(<i>b</i>) Deflection of Light by a Gravitational Field
+</h4>
+
+<div class="fig" style="width:20%;">
+<img src="images/image052.jpg" style="width:100%;" alt="image052" /><br/><br/>
+</div>
+
+<p>
+In Section XXII it has been already mentioned that according to the general
+theory of relativity, a ray of light will experience a curvature of its path
+when passing through a gravitational field, this curvature being similar to
+that experienced by the path of a body which is projected through a
+gravitational field. As a result of this theory, we should expect that a ray of
+light which is passing close to a heavenly body would be deviated towards the
+latter. For a ray of light which passes the sun at a distance of &#916;
+sun-radii from its centre, the angle of deflection (&#945;) should amount to
+</p>
+
+<div class="fig" style="width:40%;">
+<img src="images/image053.jpg" style="width:100%;" alt="image053" /><br/><br/>
+</div>
+
+<p>
+It may be added that, according to the theory, half of this deflection is
+produced by the Newtonian field of attraction of the sun, and the other half by
+the geometrical modification (“curvature”) of space caused by the
+sun.
+</p>
+
+<p>
+This result admits of an experimental test by means of the photographic
+registration of stars during a total eclipse of the sun. The only reason why we
+must wait for a total eclipse is because at every other time the atmosphere is
+so strongly illuminated by the light from the sun that the stars situated near
+the sun’s disc are invisible. The predicted effect can be seen clearly from the
+accompanying diagram. If the sun (<i>S</i>) were not present, a star which is
+practically infinitely distant would be seen in the direction <i>D</i><sub>1</sub>, as
+observed front the earth. But as a consequence of the deflection of light from
+the star by the sun, the star will be seen in the direction <i>D</i><sub>2</sub>, <i>i.e.</i>
+at a somewhat greater distance from the centre of the sun than corresponds to
+its real position.
+</p>
+
+<p>
+In practice, the question is tested in the following way. The stars in the
+neighbourhood of the sun are photographed during a solar eclipse.
+</p>
+
+<p>
+In addition, a second photograph of the same stars is taken when the sun is
+situated at another position in the sky, <i>i.e.</i> a few months earlier or
+later. As compared with the standard photograph, the positions of the stars on
+the eclipse-photograph ought to appear displaced radially outwards (away from
+the centre of the sun) by an amount corresponding to the angle <i>a</i>.
+</p>
+
+<p>
+We are indebted to the [British] Royal Society and to the Royal Astronomical
+Society for the investigation of this important deduction. Undaunted by the
+[first world] war and by difficulties of both a material and a psychological
+nature aroused by the war, these societies equipped two expeditions—to Sobral
+(Brazil), and to the island of Principe (West Africa)—and sent several of
+Britain’s most celebrated astronomers (Eddington, Cottingham, Crommelin,
+Davidson), in order to obtain photographs of the solar eclipse of 29th May,
+1919. The relative discrepancies to be expected between the stellar photographs
+obtained during the eclipse and the comparison photographs amounted to a few
+hundredths of a millimetre only. Thus great accuracy was necessary in making
+the adjustments required for the taking of the photographs, and in their
+subsequent measurement.
+</p>
+
+<p>
+The results of the measurements confirmed the theory in a thoroughly
+satisfactory manner. The rectangular components of the observed and of the
+calculated deviations of the stars (in seconds of arc) are set forth in the
+following table of results:
+</p>
+
+<div class="fig" style="width:70%;">
+<img src="images/image054.jpg" style="width:100%;" alt="image054" /><br/><br/>
+</div>
+
+<h4>
+(<i>c</i>) Displacement of Spectral Lines Towards the Red
+</h4>
+
+<p>
+In Section XXIII it has been shown that in a system <i>K&#x2032;</i> which is in rotation with
+regard to a Galileian system <i>K</i>, clocks of identical construction, and which are
+considered at rest with respect to the rotating reference-body, go at rates
+which are dependent on the positions of the clocks. We shall now examine this
+dependence quantitatively. A clock, which is situated at a distance r from the
+centre of the disc, has a velocity relative to <i>K</i> which is given by
+</p>
+
+<p class="center">
+<i>v</i> = &#969;<i>r</i>,
+</p>
+
+<p class="noindent">
+where &#969; represents the angular velocity of rotation of the disc <i>K&#x2032;</i> with respect
+to <i>K</i>. If <i>v</i><sub>0</sub>, represents the number of ticks of the clock per unit
+time (“rate” of the clock) relative to <i>K</i> when the clock is at rest,
+then the “rate” of the clock (<i>v</i>) when it is moving relative to <i>K</i> with
+a velocity <i>v</i>, but at rest with respect to the disc, will, in accordance with
+Section XII, be given by
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image055.jpg" style="width:100%;" alt="image055" /><br/><br/>
+</div>
+
+<p class="noindent">
+or with sufficient accuracy by
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image056.jpg" style="width:100%;" alt="image056" /><br/><br/>
+</div>
+
+<p class="noindent">
+This expression may also be stated in the following form:
+</p>
+
+<div class="fig" style="width:30%;">
+<img src="images/image057.jpg" style="width:100%;" alt="image057" /><br/><br/>
+</div>
+
+<p class="noindent">
+If we represent the difference of potential of the centrifugal force between
+the position of the clock and the centre of the disc by &#966;, <i>i.e.</i> the work,
+considered negatively, which must be performed on the unit of mass against the
+centrifugal force in order to transport it from the position of the clock on
+the rotating disc to the centre of the disc, then we have
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image058.jpg" style="width:100%;" alt="image058" /><br/><br/>
+</div>
+
+<p class="noindent">
+From this it follows that
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image059.jpg" style="width:100%;" alt="image059" /><br/><br/>
+</div>
+
+<p class="noindent">
+In the first place, we see from this expression that two clocks of identical
+construction will go at different rates when situated at different distances
+from the centre of the disc. This result is also valid from the standpoint of
+an observer who is rotating with the disc.
+</p>
+
+<p>
+Now, as judged from the disc, the latter is in a gravitational field of
+potential &#966;, hence the result we have obtained will hold quite generally for
+gravitational fields. Furthermore, we can regard an atom which is emitting
+spectral lines as a clock, so that the following statement will hold:
+</p>
+
+<p>
+<i>An atom absorbs or emits light of a frequency which is dependent on the
+potential of the gravitational field in which it is situated.</i>
+</p>
+
+<p>
+The frequency of an atom situated on the surface of a heavenly body will be
+somewhat less than the frequency of an atom of the same element which is
+situated in free space (or on the surface of a smaller celestial body).
+</p>
+
+<p>
+Now &#966; = – <i>K (M/r)</i>, where <i>K</i> is Newton’s constant of gravitation, and <i>M</i> is the
+mass of the heavenly body. Thus a displacement towards the red ought to take
+place for spectral lines produced at the surface of stars as compared with the
+spectral lines of the same element produced at the surface of the earth, the
+amount of this displacement being
+</p>
+
+<div class="fig" style="width:20%;">
+<img src="images/image060.jpg" style="width:100%;" alt="image060" /><br/><br/>
+</div>
+
+<p>
+For the sun, the displacement towards the red predicted by theory amounts to
+about two millionths of the wave-length. A trustworthy calculation is not
+possible in the case of the stars, because in general neither the mass <i>M</i> nor
+the radius <i>r</i> are known.
+</p>
+
+<p>
+It is an open question whether or not this effect exists, and at the present
+time (1920) astronomers are working with great zeal towards the solution. Owing
+to the smallness of the effect in the case of the sun, it is difficult to form
+an opinion as to its existence. Whereas Grebe and Bachem (Bonn), as a result of
+their own measurements and those of Evershed and Schwarzschild on the cyanogen
+bands, have placed the existence of the effect almost beyond doubt, while other
+investigators, particularly St. John, have been led to the opposite opinion in
+consequence of their measurements.
+</p>
+
+<p>
+Mean displacements of lines towards the less refrangible end of the spectrum
+are certainly revealed by statistical investigations of the fixed stars; but
+up to the present the examination of the available data does not allow of any
+definite decision being arrived at, as to whether or not these displacements
+are to be referred in reality to the effect of gravitation. The results of
+observation have been collected together, and discussed in detail from the
+standpoint of the question which has been engaging our attention here, in a
+paper by E. Freundlich entitled “Zur Prüfung der allgemeinen
+Relativitäts-Theorie” (<i>Die Naturwissenschaften</i>, 1919, No. 35, p. 520:
+Julius Springer, Berlin).
+</p>
+
+<p>
+At all events, a definite decision will be reached during the next few years.
+If the displacement of spectral lines towards the red by the gravitational
+potential does not exist, then the general theory of relativity will be
+untenable. On the other hand, if the cause of the displacement of spectral
+lines be definitely traced to the gravitational potential, then the study of
+this displacement will furnish us with important information as to the mass of
+the heavenly bodies.<a href="#linknote-27" name="linknoteref-27" id="linknoteref-27">[27]</a>
+</p>
+
+<p>
+<a name="linknote-27" id="linknote-27">
+<!--  Note --> </a>
+</p>
+<p class="footnote">
+<a href="#linknoteref-27"> [27]</a><br/> The displacement of spectral lines
+towards the red end of the spectrum was definitely established by Adams in
+1924, by observations on the dense companion of Sirius, for which the effect is
+about thirty times greater than for the Sun. R.W.L.—translator
+</p>
+
+</div><!--end chapter-->
+
+<div class="chapter">
+
+<h3><a name="chap37"></a>APPENDIX IV<br/>
+THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY<br/>
+(SUPPLEMENTARY TO SECTION XXXII)
+</h3>
+
+<p>
+Since the publication of the first edition of this little book, our knowledge
+about the structure of space in the large (“cosmological problem”)
+has had an important development, which ought to be mentioned even in a popular
+presentation of the subject.
+</p>
+
+<p>
+My original considerations on the subject were based on two hypotheses:
+</p>
+
+<p>
+(1) There exists an average density of matter in the whole of space which is
+everywhere the same and different from zero.
+</p>
+
+<p>
+(2) The magnitude (“radius”) of space is independent of time.
+</p>
+
+<p>
+Both these hypotheses proved to be consistent, according to the general theory
+of relativity, but only after a hypothetical term was added to the field
+equations, a term which was not required by the theory as such nor did it seem
+natural from a theoretical point of view (“cosmological term of the field
+equations”).
+</p>
+
+<p>
+Hypothesis (2) appeared unavoidable to me at the time, since I thought that one
+would get into bottomless speculations if one departed from it.
+</p>
+
+<p>
+However, already in the ’twenties, the Russian mathematician Friedman showed
+that a different hypothesis was natural from a purely theoretical point of
+view. He realized that it was possible to preserve hypothesis (1) without
+introducing the less natural cosmological term into the field equations of
+gravitation, if one was ready to drop hypothesis (2). Namely, the original
+field equations admit a solution in which the “world radius” depends
+on time (expanding space). In that sense one can say, according to Friedman,
+that the theory demands an expansion of space.
+</p>
+
+<p>
+A few years later Hubble showed, by a special investigation of the
+extra-galactic nebulae (“milky ways”), that the spectral lines
+emitted showed a red shift which increased regularly with the distance of the
+nebulae. This can be interpreted in regard to our present knowledge only in the
+sense of Doppler’s principle, as an expansive motion of the system of stars in
+the large—as required, according to Friedman, by the field equations of
+gravitation. Hubble’s discovery can, therefore, be considered to some extent as
+a confirmation of the theory.
+</p>
+
+<p>
+There does arise, however, a strange difficulty. The interpretation of the
+galactic line-shift discovered by Hubble as an expansion (which can hardly be
+doubted from a theoretical point of view), leads to an origin of this expansion
+which lies “only” about 10<sup>9</sup> years ago, while physical
+astronomy makes it appear likely that the development of individual stars and
+systems of stars takes considerably longer. It is in no way known how this
+incongruity is to be overcome.
+</p>
+
+<p>
+I further want to remark that the theory of expanding space, together with the
+empirical data of astronomy, permit no decision to be reached about the finite
+or infinite character of (three-dimensional) space, while the original
+“static” hypothesis of space yielded the closure (finiteness) of
+space.
+</p>
+
+<p>
+<i>K</i> = co-ordinate system
+</p>
+
+<p>
+<i>x, y</i> = two-dimensional co-ordinates
+</p>
+
+<p>
+<i>x, y, z</i> = three-dimensional co-ordinates
+</p>
+
+<p>
+<i>x, y, z, t</i> = four-dimensional co-ordinates
+</p>
+
+<p>
+<i>t</i> = time
+</p>
+
+<p>
+<i>I</i> = distance
+</p>
+
+<p>
+<i>v</i> = velocity
+</p>
+
+<p>
+<i>F</i> = force
+</p>
+
+<p>
+<i>G</i> = gravitational field
+</p>
+
+</div><!--end chapter-->
+
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author	Roger Frank <rfrank@pglaf.org>	2025-10-14 19:53:13 -0700
committer	Roger Frank <rfrank@pglaf.org>	2025-10-14 19:53:13 -0700
commit	83710a4530c12dc92fcf3e6cac9a0023b5463b48 (patch)
tree	e1db70be3fe3539571b0c4b2f324fd59a03391f4